Minnesota Care is state run health insurance. It's based on income how much you have to pay, so it may or may not be free and I don't know what sort of coverage it offers. However, having any form of insurance makes going to the doctor much cheaper.

I can't say much about University Health Care because it doesn't mean anything specific to me, but if that means the U of MN then it's usually lower cost because it's student doctors. I've gone there for teeth as I don't have dental and they're quite competent; the savings are supposed to be about 20% (their claim).