Do you want "easiest" or "most convenient"? You could buy powdered (dehydrated) eggs and just add water if you're not picky. Probably on par with the quality of eggs cooked in a microwave. I've never cooked eggs in a microwave so I can't be sure, though.

If you want like, good eggs for breakfast, I don't think you can get around using at least a pan. If you have nonstick pans, it should be pretty easy to cook the eggs and barely have to wipe down the pan with a paper towel to clean it. If you're always getting black, charred bits stuck to the pan, try turning down the heat when cooking them; it doesn't take much heat to cook an egg (unless you wanna fry it and get that crispy exterior, but even then you don't need high heat)

Guess you could go for boiling or poaching the eggs, since that involves submerging the egg in boiling water, which leaves virtually nothing to clean.

Or you could just crack a bunch of raw eggs into a glass and chug 'em Rocky-style. But raw eggs aren't as nutritious as cooked eggs so I wouldn't recommend it.