From the following verses from John we can deduce that the Sheep are the elect from both the Jews and the Gentiles:

John 10:16 And other sheep I have, which are not of this fold: them also I must bring, and they shall hear my voice; and there shall be one fold, and one shepherd.

John 10:26 But you believe not, because you are not of my sheep, as I said to you. 27 My sheep hear my voice, and I know them, and they follow me:

John 21:16 He said to him again the second time, Simon … Feed my sheep.17 He said to him the third time, Simon … Feed my sheep.

The Necessity of Prescience in John 10
John 10:27 gives us specifics as to who the sheep are. They are those who:

A) Hear Christ’s voice
B) Are known by Christ (Equivalent to Knowing Christ as well – See Gal 4:9)
C) Follow Christ

All of the above are likely ongoing since they are not conveyed in the aorist tense. It also logically follows that someone who at one point followed Christ and then didn’t follow Him later would be rejected from this description as sheep.

Since the context of the passage (especially John 10:16) indicates that these “sheep” are not yet saved but “will be” saved upon hearing the word, we must agree that the basis of their description at this point in time is clearly that of prescience (i.e. foreknowledge). Those who had not heard the Word of Christ yet (especially those of the “other fold”) were not following hearing, knowing, or following Christ yet. Therefore prescience is the only feasible explanation. Jesus knew those who would, upon hearing Him, come to know Him and follow Him. Christ foreknows the elect, because He knows the future with certainty. Though the identity of these sheep is certain, it is still contingent (i.e. depends on them hearing, knowing, and following Christ to the end). The passage at end in no way nullifies the contingencies required to be saved, nor stresses monergism. The elect are those who will hear, know, and follow Christ to the end. Paul corroborates that we are chosen “in him [Christ] before the foundation of the world” (Eph 1:4). Those who are “in Christ” are those are foreknown as believers.

Drawing and Hardening
In light of the above discussion, we can place the following substitution into John 10:26: “But you believe not, because you are not [of those who will hear me, be known by me, and follow me]”. The “because” could be less causatively rendered as “for” according to the Greek majority text. This substitution makes perfect sense. Those who whom He was addressing would never actively hear, know, or follow Christ. The general drawing by God through the Gospel will never be transformed into an efficacious drawing for them because of their pride and stubbornness. John 6:44-45 tells us, “No man can come to me, except the Father which hath sent me draw him: and I will raise him up at the last day. It is written in the prophets, And they shall be all taught of God. Every man therefore that hath heard, and hath learned of the Father, cometh unto me.” So we see that “all” will be taught of God (some larger set – In Isaiah 54 it is future ethnic Israel), but the drawing which leads us to come to Jesus is only efficacious to those who have “heard” and “learned” (a smaller subset – not mentioned in Isaiah 54). The rest are left hardened. The humble learners are illuminated with revelation, while the proud non-hearers are not (See Matthew 11:25). They will never hear and thus are not the sheep (i.e. foreknown elect) of Jesus.

The Issue of Prevenient Grace
Jesus also tells the hardened Jews,

“How can ye believe, which receive honour one of another, and seek not the honour that cometh from God only?” (5:44)

“For had ye believed Moses, ye would have believed me: for he wrote of me. But if ye believe not his writings, how shall ye believe my words?” (5:46-47)

“If any one may will to do His will, he shall know concerning the teaching, whether it is of God, or—I do speak from myself..” (7:17, YLT)

Notice the things that precede believing: seeking the honour that comes from God, believing previous revelation of Moses, willing to do the Father’s will. These passages emphasize God’s prevenient grace and the Jewish response to that grace. God gave these Jews the writings of Moses. According to His middle knowledge, Jesus claimed that if they had humbled themselves and trusted in God’s revelation via Moses, they would have trusted in Him (5:46-47). Their disbelief is rooted in their prideful search for worldly honor (5:44). Their lack of recognition of Christ as Messiah is rooted in their lack of desire to do the Father’s will (7:17). God gave the writings of Moses as a schoolmaster to lead the Jews to Christ (Gal 3:24). How gracious is our God in His Old Testament drawing. He gave the Jews enough light to acknowledge their sin, be humbled, and desire to do His will. Yet these hardened Jews resisted prevenient grace. The sheep Jesus spoke of were not such. Perhaps they already had Old Testament right-standing with God.  A careful reading of Ezekiel 34 shows a compelling case for this.  It shows that God will send David (who was already deceased - thus a reference to Jesus), and He will shepherd this flock (v 23-24).  This flock is the Spiritual remnant of Israel.  The language in Ezekiel 34 is very similar to the discussion of the sheep in John 10.  In terms of the "other fold" that Jesus references, Cornelius the “righteous” gentile (Acts 10:22) who was not “saved” (Acts 11:14) in the New Testament sense, was given the promised New Birth of Ezekiel 36:26 because of His right-standing with God. Other supporting Old Testament verses are Micah 2:12, Isa 40:11, Isa 63:11, Jer 13:17, as well as others.  Notice in all these verses that the people were sheep already because of a relationship they have with the Father.  They already belonged to the Father and would thus come to the Son (Read John 8:38-47, especially 8:47, to see how belonging to the Father was a requisite to believing the Son).  This echoes the Old Testament understanding of belonging to the Father through covenant relationship and devotion to Him (as opposed to merely being an ethnic Israelite) -- See  Ex 19:5-6, Ezek 16:8, Mal 3:16-18, Hosea 2:23.  Hence, being a sheep (i.e. belonging to the Father) was conditional on a person's response to prevenient grace.  Was the response one of humility and seeking or was it one of rejection, hardening, and covenant violation?  This response dictated whether the hearers of Jesus would believe or not.  The soil was already prepared, and Jesus foreknew the heart of each and every individual He spoke to.

Issues in John 6
First of all, we must realize that the emphasis on John 6 is belief, not election. Right in between the two parallel “I am the bread of life statements”, Jesus exclaims “Verily, verily, I say unto you, He that believeth on me hath everlasting life.” The text of John 6 in no way contradicts anything I have already addressed. Read the passage at hand:

37 All that the Father giveth me shall come to me; and him that cometh to me I will in no wise cast out. 38 For I came down from heaven, not to do mine own will, but the will of him that sent me.39 And this is the Father’s will which hath sent me, that of all which he hath given me I should lose nothing, but should raise it up again at the last day. 40 And this is the will of him that sent me, that every one which seeth the Son, and believeth on him, may have everlasting life: and I will raise him up at the last day.

These statements are directly parallel:

“All that the Father giveth me” (37) Vs. “all which he hath given” (39)
“the will of him that sent me” (38) Vs. “the Father’s will which hath sent me” (39)
“raise it up again at the last day” (39) Vs. “I will raise him up at the last day” (40)

This is therefore a passage that says something and then says it again with more detail. According to the Westcott Hort manuscripts, verse 40 starts with the conjunction gar, or “for”, which shows us that an explanation of verse 39 will come in verse 40. Therefore, we can deduce that the phrase “every one which seeth the Son, and believeth on Him” in verse 40 is the explanation of the phrase “all which he hath given me” in verse 39. Those “given” are those who continuously believe (due to the present participle, hence “is believing in him” in Young’s Literal). Those who continuously believe are the ones that are continuously coming to Christ in the second clause of verse 37 (also a present participle); They are coming in Faith. So those who are believing are also those coming (erchomai). These are “given” and therefore will come (future tense of heko in first half of verse 37) in the sense of “arrive” in the future (hence the use of a different verb here). To summarize John 6, those continuously believing are also those continuously coming. This group of perseverers in faith are “given” to the Son by the Faith and will arrive (heko). The elect are those who will persevere in faith. The elect are “given” to the Son on the basis of their faith (which is certain from eternity past – though still contingent). These “given” in John 6 are equivalent to the sheep which the Father “gave” Jesus in John 10:29, which we already established as being the elect.

Issues in John 17
In John 17:6, Jesus is praying for “the men which thou gavest me out of the world”. Due to later clarifications in 17:12 and 17:20, it is obvious that this specific part of the prayer is for the “elect” disciples only – not for all the elect. Since we established that those “given” are given through persevering faith, there is no problem here. Neither is eternal security spoken of in the sentence, “those that thou gavest me I have kept, and none of them is lost, but the son of perdition; that the scripture might be fulfilled” (17:12). First of all, the Bible makes it clear that we are “kept by the power of God through faith unto salvation” (1Pet 1:5). Secondly, John tells us how this promise was directly fulfilled in the following chapter of his gospel. He tells us, “Jesus answered, I have told you that I am he: if therefore ye seek me, let these go their way: That the saying might be fulfilled, which he spake, Of them which thou gavest me have I lost none” (18:8-9). Therefore, it is clear that physical protection was meant in His high priestly prayer.

Final Conclusions on Calvinism in John
Nowhere does John teach unconditional election. Nowhere does John teach irresistible regeneration prior to faith (See John 5:24, 20:31 – our quickening from death to life is by faith, see also 1 Pet 1:3 “hath begotten us again unto a lively hope by the resurrection of Jesus Christ from the dead” with Col 2:12-13 – “ye are risen with him through the faith of the operation of God, who hath raised him from the dead” and with Eph 2:5-6 – “we were dead in sins, hath quickened us together with Christ, (by grace ye are saved [‘through faith’ below in vv 8])”, Titus 3:5 “washing of regeneration” with Acts 15:9 “purifying their hearts by faith”). The Calvinistic idea of a pre-faith New Birth is nowhere to be found anywhere in scripture. In addition, nowhere does John teach a limited drawing (See John 12:32 “I will draw all men”), though I do agree the efficacious drawing is different than the general drawing of the Word (Jn 6:44) – Yet it is a conditional efficacious drawing and is a function of the receiver of the general drawing, not on God’s selectivity. Finally, nowhere does John teach eternal security. His teachings of security are always rooted in faith (See John 3:36 – He that “is believing” has eternal life; not He who “once believed”). For a clear teaching on apostasy, one should not belabor passages that don’t directly deal with it (such as those in John), but should turn to books that specifically address the matter, like Hebrews. Such a study should surely lead one to acknowledge the reality of apostasy from a genuine faith.