I personally distrust all non-constructive proofs as a matter of principle, and can't think of a single one I would actually miss in practice, let alone one I find convincing. They all smack of begging the question to me.

I especially distrust non-constructive proofs that involve first constructing objects in an impossible manner (like, say, sorting an infinite set of infinitely long bit strings—I'm looking at you, Cantor)—and then (ab-)using the reader's intuition about actual things you can construct to "prove" nonsense about things you clearly can't.

Which is a long way of saying that I'll only accept a constructive proof that P != NP (or vice versa).