What is the difference between Calvinist "nature" and "nature" in the sense of the Hypostatic union? In Christ, one person subsists in two natures, the Divine and the human. 

This is called the Hypostatic Union. The following quoted points explain the relevant details of what the Hypostatic Union is and what it entails. They have been obtained from a Google search, and from a major Church that accepts the Hypostatic Union.


  
  in Christ the two natures, each retaining its own properties, are united in one subsistence and one person. They are not joined in a moral or accidental union (Nestorius), nor commingled (Eutyches), and nevertheless they are substantially united. (article)
  The union in Christ is not a union of two natures directly with one another, but a union of the two in one hypostasis; thus they are distinct yet inseparable, and each acts in communion with the other. (article)
  "there is but one hypostasis [or person], which is our Lord Jesus Christ, one of the Trinity." Thus everything in Christ's human nature is to be attributed to his divine person as its proper subject, not only his miracles but also his sufferings and even his death: "He who was crucified in the flesh, our Lord Jesus Christ, is true God, Lord of glory, and one of the Holy Trinity." (CCC468)
  Obviously there can be only one infinite being, only one God. [...] God is a simple being or substance excluding every kind of composition, physical or metaphysical. (article)
  Christ possesses two wills and two natural operations, divine and human. They are not opposed to each other, but cooperate in such a way that the Word made flesh willed humanly in obedience to his Father all that he had decided divinely with the Father and the Holy Spirit for our salvation. Christ's human will "does not resist or oppose but rather submits to his divine and almighty will." (CCC475)
  the Word was joined with humanity (Col. 2:9).  Jesus' divine nature was not altered.  Also, Jesus is not merely a man who "had God within Him" nor is he a man who "manifested the God principle." He is God in flesh, second person of the Trinity. (carm.org article)
  


Note that the human nature is not absorbed or assumed into the Divine nature. Instead, it is assumed by the Divine Person of Christ. A human being is not assumed by a Divine being, a human nature is assumed by a Divine Person. 

Above I am describing certain key features of the Hypostatic Union. There is no argument that the Hypostatic Union is the correct way to see Christ. I think it is, but that is not what this question is about.



My limited understanding of the Calvinist view of the two human natures is as follows:

Calvinists believe that if we are saved, we have "two warring natures". One of these natures is the sanctified nature, and the other is the non-sanctified nature. I have heard these described as two separate wills. I have also heard of these described quite generally as two "parts". (I do not understand what it means for a nature to be a "part" of a person, except in the partless sense of the Hypostatic Union, described above).

This Calvinist view sounds familiar, because (according to the Hypostatic Union) in Christ there are two natures and two wills. But they are never "at war" in Christ: Christ's human nature always and entirely submits to His Divine nature.

Is the Calvinist use of "nature" the same as the use when talking about the human and Divine natures of Christ? If they are different, by virtue of what exactly are they different? Is what Calvinists describe some form of hypostasis, and if not, what is the relation, if any?
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.