The nominal GDP in Switzerland is actually much higher than in the US, it's just that the country is way more expensive to live in. Therefore, when adjusted for cost-of-living (PPP), the GDP are about equal.
Of course, the absolute amount of money per capita is still much higher, which lowers the relative cost of, say, medical equipment.
The private healthcare in Switzerland is also compulsory and insurers are forced to offer a basic, non-profit plan. Furthermore, individual payments may be subsidized by the government. It's therefore not that different from Obamacare, nor are private insurers left to the devices of the "free market".