I suggest you brush up on your definitions as Marxism, socialism, and Libertinism are not what you are describing here. The opposite to a socialist style health system is user pays, either out of your own pocket or the insurance company's.
You are talking economic policy not health policy. A "Health System" is a socialist policy, same as shared fire services and police forces. If the user pays for each use then that is Capitalism as a method of paying for the system, the user may not get a choice of options, but is left to pay for it by themselves at whatever cost may be. A socialist community could have a pay as you use system, the difference being its run by the community and not by state or corporation. If you have communal systems such as health and you demand a choice of opting in or out and are free to do so that is Libertarian. If the system demands you participate with no choice then that is Totalitarian and on into Fascism.
Communism is a socialist ideology so expands on socialism into more structured government, they are not one and the same thing.
So in short Socialism is a ideal of communal run services, you can run those services as opt in or out or have a number of choices for the same service. Libertarians (left wing) will demand all services and rules are optional to pay and participate in, Fascism (right wing) authority will command the opposite that the state dictates whats good for you and how you pay for it.
Capitalism is a system of economics, where the economy is allowed complete freedom to operate, which in turn means survival of the fittest. In pure capitalism there are no rules just trade and the economy sets the price for goods, until of course you get monopolies and services no one wants to do. Then you need government intervention to modify the economy to make the whole system work. Socialism and Capitalism are not directly relatable but do interfere with each other in various ways, there's no such thing as a Socialist/Capitalist slider that you can move either way in government like they make out.