I routinely go on "food trips" where I'll eat the same thing for multiple meals in a row. I just don't want anything else, only that specific food will satisfy me. Last week I had poached eggs with homemade Hollandaise sauce 3 days in a row. As long as the food is paleo I don't stress about eating the same thing for several meals in a row. Invariably, the craving goes away after three or four days.

Eggs are pretty healthy (especially if they're from real free-range/pastured hens) so I wouldn't worry about it.

Other possibilities: folate, selenium, potassium?

What I'd do is spend a bit of time typing your dietary intake into fitday.com and see if there is anything obvious you are missing on days without eggs. It's good to know what areas of nutrition you will want to improve on. And if any deficits happen to be an ingredient in eggs, then that might explain it. Beyond that, I think there are still many nutrients that are either not known or not well understood, so there may be a reason you crave the eggs that is not currently explainable by the current state of science. So just in case, I agree with others when the suggest the best thing to do is probably to just eat the eggs that you crave. Sometimes I get egg cravings, but not as often as you. With me, it's usually the yokes I crave. I'll make 5 or 6 eggs, eat the yokes and only some of the whites and then give the rest of the whites to the dog. I remember some years ago when the nutritionists starting recommending people eat the whites but not the yokes and my first instinct was HHHEEEEEEECCKKK NOOOOOOO! I ain't doing it!

Protein? Calcium? B2? Fat? Vit. A? Iron? Vit. D?