This claim is a clear bent towards Calvinism (if not even hyper-calvinism). We cannot pull isolated scriptures.  We must look at the whole Bible context. Sure God can turn a man's heart. However, to say that God turns all hearts, whether evil or good, is a clear violation of James 1:13, not to mention a very scary heretical notion -- woe to them who call evil good (Isa 5:20).

We must realize that "the king's heart" in Proverbs 21:1 is referring to one king (it's singular). Solomon was speaking of his own heart being turned by God --- As he submitted to God.

People often use Prov 16:9 in a similar manner, because they lack the diligence to do word studies such as they should:

Pr 16:9 A man’s heart deviseth his way: but the LORD directeth {Hebrew Kuwn} his steps.

Based on this verse they think God is the uncaused cause of every action of every individual (including sin). However, they fail to see the same Hebrew verb used 6 verses earlier which shows that this "directing, establishing, turning" is conditional of "commiting thy works unto the Lord". This shows a synergism between God and man.

Pr 16:3 Commit thy works unto the LORD, and thy thoughts shall be established (Hebrew Kuwn)

In the same passage you referenced (prov 21), it clearly states that God "directs" the upright:

Pr 21:29 A wicked man hardeneth his face: but as for the upright, he directeth {Hebrew kuwn} his way

God will direct a humble believer who seeks His will:

Psa 10:17 LORD, thou hast heard the desire of the humble: thou wilt prepare {Hebrew kuwn} their heart, thou wilt cause thine ear to hear:

God will direct the righteous man in whom He is delighted:

Psa 37:23 The steps of a good man are ordered {Hebrew kuwn} by the LORD: and he delighteth in his way.

Israel was not directed by Him in the wilderness:

Psa 78:37 For their heart was not right {Hebrew kuwn} with him, neither were they stedfast in his covenant.

Therefore, to be directed by God requires your trust and acknowledgment of His ways:

Prov 3:5 Trust in the LORD with all thine heart; and lean not unto thine own understanding. 6 In all thy ways acknowledge him, and he shall direct thy paths.

Rom 8:14 For as many as are led by the Spirit of God, they are the sons of God.

CAN GOD TURN ANY HEART? YES. DOES GOD TURN EVERY HEART? NO