Well if you're selling them you shouldn't have to worry too much about where to keep them, until you manage to sell them though you can store some of them in the PC. They won't hatch unless you're carrying them.
Some people leave them with the Breeding Center too, I think.
I haven't tried that but I think you can either sell or donate them to the Breeding Center.
You'd probably make more money selling them on your own but if you drop them at the Breeding Center it's probably quicker, since they pay you for all the eggs at once.
I prefer to know where the eggs are going myself, so I don't use the Breeding Center for that either. I think people are usually pretty interested in eggs anyways, I haven't had too much trouble.