The answers will vary based on the country you are talking about.
Not every country promises a "right" to healthcare. Not every country allows its people to choose conditions under which they work. That is an important caveat to state up front.
Most people who talk about "healthcare as a right", at least in the United States, are making an empty rhetorical flourish.
That is, to say "healthcare should be a right" is really a highly moralistic way of saying "I think healthcare is very important." It's hard to discuss such statements seriously in the context of your question, since the entire point of this construction is to avoid discussing the logistics and economics of actually supplying everyone with healthcare that is "good enough." People who say this don't really want to get into a discussion about economics; they want you to agree with them that they are good people for caring about others. You really shouldn't attempt to enter into a serious analysis of such statements unless the person also is presenting a policy proposal that has real detail. You'll just end up making the other person confused and angry.
That said, I think the conflict here is worth talking about anyway, so we should start by clearing up some terminology.
What is a "right" anyway?
There are actually two kinds of rights relevant to your question. Positive rights are those which assert that you have the right to be provided something, like in the United States, how everyone is entitled to legal representation in a criminal trial. Negative rights are those which assert that you are not subject to action taken by someone else, such as the right not to be prevented from practicing the religion of your choice (including none at all). Sometimes, positive rights are also known as entitlements, and negative rights are also known as liberties.
The combination of entitlements and liberties within any given society are not guaranteed to create an economically optimal outcome, and often do not.
Your question presumes implicitly that, if we assume that there is some serious entitlement for a good or service that conflicts with liberties of people in society, that someone has noticed this and would fix it in some way in some economically beneficial fashion, e.g. to produce "good enough" healthcare for everybody in this case. This almost never actually happens, because this business of liberties and entitlements is actually a superposition that emerges from competing concerns within society going back centuries, almost none of which have anything to do with optimal socio-economic outcomes. Such superpositions emerge and become stable because of how differences in values between various people get resolved. Values are at best difficult to assess objectively for any non-trivial moral question.
So what ends up happening typically is that the values of the society are what they are, and then the economic outcome is the result of objective reality not being optional. If people don't like that outcome, they try to add entitlements or take away liberties to address it. Either one of those is hard to do, so there's actually no guarantee that it works out.