Bots do not lay eggs on the rump. I've seen them lay eggs low along the belly line and in the flank area where horses itch themselves, but not high on the rump.
If those are truly eggs you're looking at, they are most likely "nits", as in lice nits
But it's hard to say without pics. It might be vet time, to confirm whether or not the horse has lice nits (which is contagious) or something else.
The vet can also provide or recommend the necessary shampoo or wash to get rid of The Garbage
Reading/replying to this thread is not going well with my oatmeal - lollol