Both. As you say they contain both protein and fat. Am I misunderstanding the question?

I got a lot happier about eating when I stopped looking at food as a macro-nutrient breakdown and instead started to look at it as food.

Instead of thinking about the breakdown of my food I have shifted instead to thinking about why my food was made and who made it. I try to limit myself to eating things where I know more about this and feel more comfortable with the motivations.

An egg is an egg, it is layed by a hen, unfertilized usually, and backed with stuff for growing baby chickens.

That is a motivation I can get behind. Sure it is better if the person caring for the chickens lets them be chickens and eat bugs and forage, but the chicken is the primary producer and makes eggs to nourish, there is a limit as to how much that can be messed up :)

Lots of foods have lots of more than one macronutrient.

Because foods contain a combination of fats and proteins, as well as a combination of types of fats, it can get confusing. For instance, the Jaminets suggest 15% of your calories come from protein. At first glance, I thought that meant the total calories from meat and eggs. But that is not true. The meat and eggs are also part of their 65% fat recommendation. What I do now is just eat following the 3 main guidelines of no grains, sugars, or seed oils w/ only a little full fat dairy. I eat when I am hungry and what I want to eat. I enter it into my software and then I can see the breakdown of my nutrients. Over time, I can see what I am averaging for the week, month, or even year. I probably won't keep this up, but for now it gives me data I can use. It is easier for me to do it this way than to plan out menus the other way around. Some days I am just downright hungry and others not so much.

Egg fat-wise: don't forget the Omega 3 EFAs in the egg yolk! (These die when you cook it though, so I usually separate out the yolks and put them in raw over my cooked food. Yum!)