I'm a U.S. native. This is another one of those amazing "I-know-but-I-can't-say-why" questions.
To me "bird egg" sounds incorrect and "bird's egg" sounds correct, _despite_ the fact that "chicken egg" sounds correct.
I think the reason is that to me, "chicken" is acceptable as a mass noun; we're having chicken for dinner. What kind of meat is that? it's chicken. They have chicken breasts on sale at the supermarket today. What do frogs' legs taste like? They taste like chicken.
However, we do not use "bird" that way. Can you say "we're having bird for dinner?" Can you say "rattlesnake meat tastes like bird?" I don't think so.
You cannot use "bird" as a synonym for "poultry." "He raises poultry?" Fine. "He raises bird?" No.
All that said, the apostrophe-s possessive is disappearing rapidly. I live in a state where my driver's license says "driver's license" on it, but I think in most states, they say "driver license."
And, oh dear... forgive the vulgarity... although "bird egg" sounds wrong to me, "bird [poop]" sounds correct, and the two words can even be joined to form a compound word.
I guess I don't know.