That's not true by the definition of "nationalization" (government ownership)
Switzerland and Netherlands are entirely private insurance. Most European countries have a public system and private system. It's very common for people to purchase supplemental insurance on top of what the government provides.
Prices do vary between companies but many people just use the official insurance comparison tool and choose the cheapest as the services are completely identical. I'm completely against that system and would prefer a national health insurance like Japan.