To me, "on account of him" sounds very strange. Usually it is "on [my/his/her] account," For example. "Please, do not stop on my account," It might be okay both ways but I feel the first is very odd. I'd rather say, "Because of him, we do not have food," instead of "on account of him, we have no food," as it sounds better to me but still could be correct.

The context could change the meaning. For example: - Don't stop on account of him == Don't stop because of him - Don't stop on his account - could, in some contexts, mean don't stop working on his (bank/sales/customer) account. There is room for ambiguity here. This might be a US/UK English thing, but for me (UK) I find 'on account of' far more natural, as opposed to Shane.