Considering the Pokemon Center seems more than happy to store them for me, I don't even have to carry them around at all. The Breeding Center seems more hassle than it's worth.

[ Also he likes keeping tabs on his eggs. ]

And I'd imagine they'd simply find a way to avoid the camera's view. Even if you think all points of a room are covered, they might be able to sneak into a corner or hide behind something. Or simply destroy the cameras and be done with it.