No they don't:

If you work, you automatically get enrolled in a hybrid healthcare system (state+private)

If you are young or elderly you get automatic coverage.

If you are working age but have no job --then you get basic healthcare (not comprehensive)

Lots of people who can afford will buy private insurance on top if they have a serious illness (basic healthcare is not comprehensive)

Also copays and high deductibles. People can end up footing 20 to 50% of their medical bills (out of pocket).