They were separate nations, and Eggs lived in a completely different universe very far away. They could've been racist to them, and had some Eggs as slaves tho. I give you permission to make an Egg OC or two that's ruled by the bird folk.
There is a seventh species in the Inkverse. Birdfolk; A Path I Can't Follow (Revenge of The Sith); "I have brought Peace, Freedom, Justice, and security to my new empire!"
"Your new empire Anakin my allegiance is to the Republic, TO DEMOCRACY!"
"If you aren't with me then you are my enemy"
"Only a sith deals in absolutes, I will do what I must"
"You will try"