You don't jam your hand into a hen's ass to extract eggs.
Technically, birds have cloacas, so it would be the right oriface to go fishing for such things.
Regardless, while it may be more realistic for the eggs to just fall out of a chicken (preferably a non-caged chicken that needs to graze, or at least a caged chicken that needs food, so that you're not just cramming 50 chickens into a single cage that spontaniously overflows with eggs with no upkeep), making the farmer heft the chicken, take it to the farmer's workplace, hold the chicken in the right way and shake it until the egg falls out would at least let it use the same streamlined mechanic that would let milking, shearing, and egg-gathering a fairly quick and easy coding job that wouldn't be too much to ask of Toady.
Hamfisting in new animals or arbitrarily limiting dwarves to only a portion of the livestock variety would add little in the way of flavor.
I'm not saying we bar dwarves from having dogs. I'm saying that we already have subterranean ecologies with subterranean crops, and it only makes sense that a subterranean race has livestock adapted for subterranean life.
Besides, it would be fairly unique to offer an alternative to cats with geckos that whip vermin with their sticky tongues, alternative dogs with cave boars, and maybe a giant mole or a land worm or something for an underground pack animal.