There are three obvious advantages.
You can trust that the level of coverage won't change. Public health insurance policyholders have to accept that their coverage may be restricted at any time by the state. The coverage that is listed in your policy when your insurance kicks in is guaranteed for life.
Private health insurance is fair to all generations. You use your old-age provisions to finance your own healthcare costs in old age, and thus do not burden the next generation.
In many cases, private health insurance is cheaper than public health insurance, despite offering better coverage.