Will self funding a PhD hurt employment chances?
There's nothing on your CV that needs to indicate exactly how you were supported during your Ph.D. In fact, if you were self-funded, and managed to get an occasional fellowship or other form of support, that's a plus !

All a recruiting committee should (and does) care about post-PhD is the quality of your work (for faculty positions there are additional issues). No one cares about how you were paid to do that work.