Christian existentialism has no particular affinity with Calvinism or the Reformed Tradition. Of the eight major Christian existentialists (Kierkegaard, Dostoevsky, Marcel, Jaspers, Barth, Tillich, and the Niebuhr brothers), only Barth comes from a Reformed background. (Kierkegaard and Tillich were Lutherans; Dostoevsky was Russian Orthodox; Marcel and Jaspers were Catholic; and the Niebuhrs were members of the no-longer extant German Evangelical Synod.) It's not that hard to see why, I suppose: Existentialism involves the ultimate freedom of man, and Calvinism is famously caricatured as "that predestination denomination"--which it is, at least to some extent.
But I'm an existentialist for the same reason I'm a Presbyterian, which is that Karl Barth was and I agree with him on just about all of the major issues. (And I did before I'd ever read him, thanks to the popularization of his theology in Frederick Buechner's books.) And so there must be a synthesis (a bad word for Barth, of course) here; there must be a way to believe in predestination and the ultimate freedom of man.
I found it on my third reading of Church Dogmatics: A Selection. (I still have neither the money nor the guts to read the Dogmatics in their entirety, which I suppose makes me disingenuous when I say that I agree with Barth on just about all of the major issues.) Mankind, as a creature, has a special type of freedom, a freedom which God the Creator does not have: the freedom of alienation. This alienation manifests itself, at least when I talk about it, in three ways: alienation from oneself, alienation from one's fellow-creatures, and alienation from God. God cannot, by His very nature, be alienated:
it is a mark of created being as distinct from divine that in it conflict with God and therefore mortal conflict with itself is not ruled out, but is a definite possibility even if it is only the impossible possibility, the possibility of self-annulment and therefore its own destruction.This "impossible possible," this possibility for alienation and indeed annihilation, is what our freedom would look like if we held it up to the light. But this freedom is indeed ours, and we are able (and most of the time, willing and ready) to exercise it.
But Barth is not Heidegger. A mere exercise of our freedom is not sufficient to end alienation--indeed, most if not all expressions of human freedom lead only to further alienation. (For a great example, head over to John Updike's Rabbit, Run, which the author has said he fashioned as an object lesson from Barth and Kierkegaard.) We have the ability to choose, but because we are (as Plato says) fundamentally stupid and (as Calvin says) fundamentally perverse, we will choose a Tower of Babel over the Living God every single time. Our freedom leads not to Being but to Nothingness.
The solution, then, is to recognize that our freedom is real but still illusory--that is, we are free but it's not actually a good thing to act on that freedom. (In this sense, our freedom is that of Adam in the Garden; we can choose to eat from the tree or not to eat from it.) Our freedom is an attempt to be like God, and indeed, "From the very first man as such has continual illusions about himself. He wants to be more than a creature." The problem is that he isn't more than a creature.
And so it is that the Christian must relinquish his freedom. The Christian, according to Barth, is defined by his acceptance of his own contingency: "Of all creatures the Christian is the one which not merely is a creature, but actually says yes to being a creature." He submits himself to God's sovereignty (which Barth, good Calvinist that he is, believes will take precedence in the end anyway) and recognizes that this is the dignified and proper place for him, that "it is the glory of the creature to be lowly in relation to God."
Now, I recognize that this in no way solves the age-old problem of the connection between man's free will and God's sovereignty, but it does explain how the Christian existentialist can believe in ultimate freedom and ultimate sovereignty. You just can't put them on the same level. You end up like the canaries at the top of this post. You're able to leave the cage, but you don't--because you recognize that the cage is not a place of imprisonment but the place you belong.