My cleaners molt quite a bit, but I have never seen them leave eggs behind. As a matter of fact, the eggs appear to be green in color, so I'm really not sure what that is on yours. I always pick the molted exoskeleton out and throw it away, but I've heard plenty of people say they leave them in the tank, so it probably wouldn't hurt to leave it in there. On the other hand, even when my shrimp's eggs hatch, they never last very long before they're eaten by the tanks other inhabitants, so unless you're planning on removing them to a separate breeder tank, don't expect any of your baby cleaners to grow up.
I guess that's my very long-winded way of saying that I'd chuck the thing.