What is the difference between Calvinist "nature" and "nature" in the sense of the Hypostatic union?

The way Calvin used the terminology of nature is completely different than what Chalcedon refer to with nature. For Calvin the unity between the Logos and His humanity is akin to psychosomatic unity between human's soul and body. This is why R C Sproul stated that God can't die, the one died on the Cross was His humanity not the Logos. For Chalcedon the unity is between a divine person with His humanity. Why this is different than that of Calvin? Because His humanity belongs to the Logos, the only subject who suffer was a divine person. Nature can't die, a person does. St. Leo the Great can confess that God died on the Cross. While Calvin can't. Because for him the Logos is not the only subject of Christ.

All are agreed, however, that the Creed affirms that the two natures exist in one person. However, the debate’s major point of contention was the identity of the one person. In other words, do we simply identify the person with the divine Logos (Cyril’s position) or with the whole Christ (Calvin’s position)?

Beeke and Jones, A Puritan Theology, p. 337.

Calvin also teach that on the Cross there was real chasm between the Father and the Son. The Son was damned to be burned in Hell for us. This show how he distinguished the Logos and His humanity. For Calvin the Logos can't die and suffer in Hell but His humanity can.

But we must seek a surer explanation, apart from the Creed, of Christ's descent into hell. The explanation given to us in God's Word is not only holy and pious, but also full of wonderful consolation. If Christ had died only a bodily death, it would have been ineffectual. No, it was expedient at the same time for Him to undergo the severity of God's vengeance, to appease His wrath and satisfy His just judgment. For this reason He must grapple hand in hand with the armies of hell and the dread of everlasting death.... No wonder, then, if He is said to have descended into hell, for He suffered the death that God in His wrath had inflicted upon the wicked! ...He paid a greater and more excellent price in suffering in His soul the terrible torments of a condemned and forsaken man.

Calvin, Institutes, 2:16:10

Most simply put, when the Calvinist uses the term "nature" in this context, they are not using it in the same way that they are using the word "Nature" in terms of the Hypostatic union. The Calvinist is really more using this as a way to say "tendency" or "leaning" or even "will." Human beings will only ever have one Nature (in the Ousia sense of Chalcedonian Christology).

It's a good reminder that we should all be careful how we use terms.

Regeneration, initiated by new birth, is the Evangelical belief that when a person believes in Christ they are mystically united into Christ. Their old nature dies in him and a new nature is born. Rebirth, or regeneration is when the image of God is restored within a sinner. Calvin thought of it more in the fullest sense of continuing into sanctification and resurrection of the body. Typically Protestants in terms of the doctrinal mechanics of regeneration identify is as the inward renewal beginning with justification by faith and continuing through the process of sanctification.

The two natures we are talking about are simply 'ourselves' sanctified and those aspects of ourselves not sanctified. These two parts of ourselves war against each other. That is why, for example, Paul say's:

So I say, walk by the Spirit, and you will not gratify the desires of the flesh. For the flesh desires what is contrary to the Spirit, and the Spirit what is contrary to the flesh. They are in conflict with each other, so that you are not to do whatever you want. (Galatians 5:16-17, NIV)

In a completely absolutely different sense, a human being was assumed into a divine being, so that two fully different natures subsist in the incarnation. There is no parallel of the incarnation with sanctification, so the question seems to be an identification of a gross ignorance of regeneration. It is actually equivalent to a question someone asked Jesus:

You should not be surprised at my saying, ‘You must be born again. ’ The wind blows wherever it pleases. You hear its sound, but you cannot tell where it comes from or where it is going. So it is with everyone born of the Spirit.” “How can this be?” Nicodemus asked. “You are Israel’s teacher,” said Jesus, “and do you not understand these things? (John 3:7-10, NIV)

The best way I can explain it to a person that considers water baptism as the initiation into God's kingdom, rather then personal faith, is to say: 'Imagine everything you think that happens in water baptism happens when a person believes in Christ.' The resulting two natures, sanctified and not sanctified, is the Protestant understanding of the two natures after regeneration initiated by faith in Christ.