Anything that requires concessions on the part of another individual in any form, including money, labor, material or property of any kind - cannot be considered a "RIGHT".
Healthcare requires money and/or labor of another individual. You can't make anyone do anything. Therefore healthcare can never legitimately be considered a right. It is a SERVICE. And it you get it for labor it is a BENEFIT. And if you ever get it for free it is a PRIVILEGE.
That's only if you focus on the short term. In the long term, a healthy population benefits everyone - including business owners - through increased productivity, fewer sick days, less chance of infectious diseases spreading to you or the ones you love, and so on. If some people get healthcare and others don't, those that don't create a drag on the economy that hurts everybody - not to mention that it negates the large-scale benefits listed above. Since everyone benefits from everyone getting preventive healthcare, but the market can't correctly price that benefit (called a "market failure"), the only way to ensure that everyone gets healthcare - and that the benefits to everyone (including you) are fully realized - is to make it a public good.
As to your argument that "you can't make anyone do anything," well, that's true as far as it goes. However, as sinwin said:
Money made by those who can afford healthcare is made at the concession of those providing that money, or the labor that earns that money.
So, you don't have a
right to the money that you use to pay for your personal healthcare either. You only have that money because of the goodwill of other people, and it's not unreasonable for those other people to expect you to contribute your fair share to something that benefits everyone; without their goodwill, you'd be destitute.