Draw when heads up?

[schwza]

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(I don't believe any fold equity exists) Is it worth a call for all your chips when playing for 1st and 2nd when you "know" that you are behind?

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it's not a call for all your chips - it would be a push for all your chips. if you're 100% positive that he'll call a push and he'll have a Q, it's close enough that you could go either way. except, if you're 100% sure that he has a Q just because he led out after calling your raise, then he's obviously completely terrible or you're psychic and you should avoid playing any huge pots where you're not a big favorite.

calling may still be better than folding, depending on how often he pays you off if a Q/A hits, and whether you can push him out if some other scare card (like a K) hits.