I eat a lot of egg whites daily too, usually 6 for breakfast and another 4-6 hardboiled whites in the afternoon. I also eat chicken everyday for lunch (boring), a red meat, fish or turkey for dinner, drink 2 cups of skim in my morning shake and supplement with whey & casein everday.
Like the others have said, a mix of protein sources gives you the right amino acid profile. A single protein source can deprive you of certain amino's. that being said, egg protein is pretty much a grand slam in terms of amino profile. Think about it, it's there as food for another mammal to develop and grow. Eggs have all the essential amino's, BCAA's, glutamine, and non-essential amino's. It's literally got it all. BUT, I'd still recommend a balance of protein sources, just like the others here said.