Technically, being a former patient at the facility should not affect whether you could ever get hired there--if that was the case, there'd be many nurses ineligible to work in a lot of places
Reality: depends if you're OK with working there AND if management would be OK with your working there. Management may not feel comfortable having you on that side of the fence because you're going to have a dual relationship with them: as staff and as patient.
Also, HIPAA can protect your PHI, but HIPAA doesn't protect against a patient
seeing you and saying to their friends/family, "hey, the nurse there is also a patient of Dr. X!" So some of your privacy can be at risk, especially if you keep running into the same people in the waiting room as you go to your own appointments.
Personally, I wouldn't do it.
Best of luck whatever you decide!