@scott
" chickens' egg "? Surely not!
Doesn't this describe a single egg which was laid by more than one chicken? A physical impossibility, I would have thought.
Hopefully, it was a typo; if I'm wrong I will certainly have egg on my face!
I would write:
" the chickens' eggs " - the eggs from a specific group of chickens
or " the chicken's egg(s) " - the egg(s) from one specific chicken
or " chicken egg(s) " - where 'chicken' qualifies the egg type, ie eggs that came from chickens, rather than from ducks, geese, dinosaurs, alligators, etc.
I suppose that, in this particular case, the phrase " the chickens' egg " is probably not grammatically incorrect, merely improbable, as it is theoretically possible for a group of chickens to have possessive claim to ownership of a single egg in an anthropomorphic sense. However , confusion arises because " chickens' eggs " or " a chicken's egg " implies that the egg 'comes from', rather than 'belonging to' chicken(s). If you just want to describe the egg(s), then it's probably better to avoid the apostrophe dilemma altogether and simply use " chicken egg(s) ". Or " hens' eggs ". Uh-oh, another apostrophe!