"The white part of the egg" is shortened to "the white of the egg" which is shortened to "the egg white" (you can think of this as "the egg's white", not that anyone actually says that out loud). :) And so we're talking about a thing called a "white", so that's the bit we pluralize. In "egg white", "egg" is a modifier, so that's not the part you pluralize. There's also "the white of the eye" (as in, "don't shoot until you can see the whites of their eyes". Hence, again, you pluralize "white". But no-one says "the eye white" (which you'd expect from "egg white"). I don't claim that this is a foolproof way of thinking about this. There might be irregularity than I mention. :)

egg whites

no it's egg whites