I could see it going either way.
With free access, people would be more inclined to go to the doctor for simple and small things, but in return would probably catch more serious issues early and have better access to treatment, therefor reducing the need for intensive and specialized healthcare.
Without, people avoid going to the doctor for small stuff, but end up having to go in with more complicated issues later on.
There was a study years ago about putting up Jersey walls on parts of US interstate highways, to see if they increased or decreased the number and severity of accidents. The conclusion was that it increased the number of minor accidents, and decreased the number of serious accidents.
My guess is that universal healthcare would be the same: increasing the number of minor visits and (by noticing and addressing issues before they became serious) decreasing the number of serious visits.