The classification of eggs as meat, dairy, or other depends on the classification system that you are applying. For religious purposes, eggs can be classified as either depending on the religion. Technically, they are neither.
A vegan would consider it off limits. I'd classify eggs as dairy for convenience even though it sounds silly. As a low-fat vegetarian, it's OK to cook with egg whites, but I personally worry about the eggs that are imported from China. For safety sake, I refuse dairy and I refuse eggs.
I lived in Michigan when the dairy milk was tainted, resulting in a huge number of serious illnesses. Milk is often contaminated with known and unknown chemicals and hormones. The same is true of eggs. If you know the right food combinations, a low-fat, low-processed-carbohydrate vegan diet is much more healthy than a typical American high fat, low fiber diet. It seems like groups like the dairy counsel will promote whatever fattens the wallets of their members.