Well, I work with eggs everyday at work. I work in the dairy department of my grocery store. Thats what I'm Sham Sham the Dairy Man. Anyway, I have eggs break and dry in the egg case all the time. It gets hard, sticky, and nasty. Warm/hot water will turn that hard mess into something softer to work with. Do this till there isn't any physical egg left. There will probably be areas of visable imperfections. I'd suggest 3m Bug and Tar remover. I detail cars on the side, and this stuff kicks ass! Its safe on paint, but very strong and effective. It will take off any wax, but is clearcoat safe.

I haven't cleaned up egg off a car, but the above info is coming from expereince with eggs and cars.