They both work fine, depending on how you meant to structure the sentence.

"We'll all be eating rabbit" and "We'll all be eating rabbits" have slightly different connotations (generic term for the meat vs specific animals) but mean effectively the same thing. The former seems to be the more common usage, as it's more analogous to "We'll all be eating beef" instead of "We'll all be eating cows".

I'd say the jury/consensus is still out, or rather, hasn't even been convened. One could also conceive of it as a mass noun in the context of consumption, similar to corn, beef, or lettuce.

If you are talking about the food substance itself, Cactus is correct.