"You eat egg", although a little strange sounding to this American, would represent that you're eating the substance called egg, without reference to how many literal eggs the egg substance came from.
Cf. "You eat chicken" vs "You eat a chicken," or "You eat beef," versus "You eat a cow."
(As you probably know, "I had egg on my face," is an idiomatic way to say I was embarrassed. In that phrase egg refers to the non-count substance, not a countable object; presumably by the time you get it on your face, it can't be traced to one particular egg.)