I live in Canada, with a single payer health care system, and it isn't free. I pay for it with my taxes every year. Because everyone above poverty line pays into it, costs are low to me. The government forces me to pay it, like the local government makes me pay my property tax. This way, everyone gets to use the services that were purchased using our pooled resources. When you insure your car, you benefit from other payees when you total your car. Why is health care any different?
Well, for starters, because your personal health is not a public good that is being used by other people, the way most of the things paid for by property tax, for example, are. And car insurance which covers YOUR car in the event of an accident is voluntarily paid for by the car owners. In the US, at least, the car insurance which is mandated by the government is to protect OTHER people's property from your stupidity. You're perfectly welcome to leave your own car's damaged uncovered by insurance if you want to, so long as it's not owned by someone other than you, who is then going to suffer financially for your stupidity.