If the hospital/provider sends your bill to a collections agency, then it can definitely show up on your reports. Especially so if you are actually sued for the debt, in which case the judgement is also a public record.

I've had this happen a couple times in the past when I was in treatment for cancer and underemployed. One agency reported the collections action and it went on my credit report (no indication that it's medical debt or anything else, so I imagine it would be up to the consumer to contest these things with the bureau?) Another collector didn't, so I never paid the bill or heard from them again!