No, health care in Sweden is not solely financed through regional income tax. Large parts of the social security system are financed through payroll taxes and corresponding mandatory contributions from self-employed people (the latter is called "egenavgift" -- feel free to Google it).

But looking again, he's in fact born before 1983, which means that he has to pay pension contributions as well, bumping the total contribution up to nearly 30%. In addition to income tax, that is.