Governments already provide "free income" in the form of free or subsidized services.
Say you have a fire-department even though you personally might not be paying anything for it because you are so poor that you don't pay any taxes. You have police protecting you and the army. You have free primary school at least.
So I think the question is, would it help for the government to provide more, or less, or the same amount of free services as it does currently?
Would it "increase prices" if healthcare was free? Not necessarily I think. At least not the price of healthcare. Government would be in a much better position to negotiate drug-prices with pharmaceutical companies, than individuals are.