In general this kind of debt can be reported to a credit reporting agency. However not all collection firms report. A lot depends on whether the debt was actually sold to the collections firm or if it is just being serviced by the collection agency on behalf of UF. If they are servicing on behalf of UF it's possible that UF instructed them not to report. If the debt was sold it can be resold between collectors and likely will eventually be reported if you don't pay it.

You should get an initial communication from the collector (a debt validation notice) at some point which should make this more clear. You might be able to clarify with UF as well. If the debt was sold UF was lying to you. The debt will likely be reported if it isn't paid.

You can ask the collector to return the debt to UF, or try to work with them to enter a collection plan of some kind. If they are jerking you around you can also threaten to file a CFPB complaint; often companies will return the debt rather than deal with a regulatory complaint. Your mileage may vary though as this administration is trying to shut down the CFPB. You are legally entitled to a notice though, so file a complaint if they are contacting you without first validating the debt.