As someone who has had my medical chart accessed by someone working at the hospital (i.e. HIPAA violation, and yes, the person henceforth had some privileges quietly revoked but was not formally reprimanded) I, personally, do not think you are being paranoid. HIPAA violations happen. But I agree with others in that A) Why work for a place that would do that or punish you for seeking medical care and B) You'd have legal recourse if they allowed your medical record to impact employment (if you could prove the link). I, personally, do in fact seek out a separate hospital system for personal care than the one I've volunteered at. In fact, I recently had a (personally) embarrassing situation and specifically asked the EMTs to go to the hospital where I am not a volunteer and shadow (they are equidistant). I plan to keep my care separate from wherever I end up employed, as well. Again, just a 2 cents from someone who is often called paranoid but sleeps better at night for it.
*That being said, I would never turn down a great opportunity for something like this. If you want this job, take the chance. Odds are incredibly low that anyone would access your record, and even lower that they would discriminate you for it.