Chickens are an old-world animal, domesticated from the red junglefowl of Southeast Asia. IIRC there's some possible evidence for chickens being taken to South America by the Polynesians, but they certainly didn't become widespread in the Americas until the European colonizers showed up.

Maybe Native Americans ate the eggs of other birds that they did have access to, such as turkeys? But even if they did, it's chicken eggs that are the ones easily commercially available in the quantities a restaurant would need, so...