This reflects my experience in Canada. It's really hard to figure out how much of your taxes goes specifically towards health care, but we put a significant amount of effort into a calculation a few years ago and came up with a number in the same ballpark as the premiums for a very good private plan in the US. The problem is then trying to compare the composition; there are lots of aspects of the Canadian public plan that are superior to the USA, notably access to frontline services and certain widely used drugs (in the news: cheap insulin w/o a prescription). There are also many components that are vastly inferior, like timely access to diagnostics or procedures that seem "quality of life" related but really are not (ex: delayed hip replacements lead to major health issues).

So it's not just free vs. pay, but also what exactly you're buying. Unfortunately the single public payer in Canada system is a sacred cow, much like the idea of "socialized medicine" in the US is a non-starter. Everyone just digs their heels in and holds to the extremes, so the idea of small experiments of change never gets much traction...