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.
Costs can go up or down overall, as more demand is created. It all comes down to what will be covered. Cost per treatment or cost per service should drop significantly.
The USA already spends way more than most other countries for worse outcomes.