Went to bathroom in public toliet with flies or nats in stall, can they cause infection if come in contact with vaginal/anal area?

Casual contact- no: When toileting, the casual contact of a fly on the vaginal anal area won't cause an infection since these areas already contain a high concentration of normal human bacteria (flora) that will overwhelm whatever little disease-causing bacteria the fly deposits from their feet. Of course, all bets are off if you forget to wash your hands after toileting- so wash you hands!