I don't think that I'd count what was said as resignation either, as it was clearly a reaction to a perceived checkmate.  If some other move had been played (such as capturing hanging piece) and a player explaimed "argghh, 0-1 for them" I might think of that as resignation, but not when the person thought they had just lost due to checkmate.   

I don't know blitz rules very well, though.

Whatever about declaring a loss aloud, I would be a little scared of anyone who said that sighing acknowledged defeat.

Sorry, yes, missed that part! Hm, in blitz. Well, in that case you should have one I guess, since he pressed the clock after an incorrect move.

And normally, isn't both players supposed to sign the score sheets in agreement? In this case, I wouldn't count your statement as a resignation and since you hadn't signed the score sheet, I would think a reasonable referee would ask for the same thing as Tony asked, i.e. go back to the game after Bxh7+.

As ...Qxh2 was illegal, why didn't you both go back to the previous position (after Bxh7+) and continue the game? Or was this blitz tournament really so serious?
I know it's possible to get a bit carried away in the heat of the moment, but here the game has barely started.

Not entirely sure on this, just to get the rules in the thread:

http://www.fide.com/component/handbook/?id=124&view=article

B.3.c. An illegal move is completed once the opponent’s clock has been started. The opponent is entitled to claim a win before he has made his own move. However, if the opponent cannot checkmate the player’s king by any possible series of legal moves, then the claimant is entitled to claim a draw before he has made his own move. Once the opponent has made his own move, an illegal move cannot be corrected unless mutually agreed without intervention of an arbiter.

5.1.
a. The game is won by the player who has checkmated his opponent’s king. This immediately ends the game, provided that the move producing the checkmate position was a legal move.
     
b. The game is won by the player whose opponent declares he resigns. This immediately ends the game.

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The way I interpret the rules (mind you, I'm not an arbiter):
- 5.1.a. doesn't apply (not a legal move)
- You didn't do your own move, so you'd be able to claim (no time limit at all)
- 5.1.b. MIGHT apply - if your statement is the same as a resignation
- There is no way to claim after you have resigned at all (not 3 seconds after the resignation, either), as the game has ended

The question would now be whether "argh, 0-1 for them." counts as a formal resignation.

I would say no, as 
1) This isn't talking to your opponent at all, but rather to your teammates (could also say this as a joke after the opponent plays 1.e4, or whatever - I doubt anyone would register that as a resignation); 
2) The only thing I found regarding resignation procedure in the rules was "The game is won by the player whose opponent declares he resigns.", and your statement seems to me like less of a 'Declaration of Resignation' and more of an 'Expression of Shock' (like shouting "Oh, damn!" as your opponent sacrifices a rook on e6, which you completely overlooked)

Again, I'm not an arbiter, this just looks like the most logical interpretation of the rules to me.

FIDE or USCF (or other) rules? I'm not sure it would make a difference, but it's good to nail down the context.