I purchased a Park Tool kit (the AK-37 to be exact) that came with their version of the chain cleaner. I have never used it: I always found it easier to remove the chain and clean it in solvent or replace the chain altogether, they aren't [I]that[I] expensive.
In response to your question though, unless you use an inordinate amount of degreaser and spill it all over the rear hub area, I would not worry about it hurting anything. Good luck!
Cheers
lverhagen