I personally find it easier to sidestep the rights issue and just say "we CAN ensure everyone has healthcare, so we should do that". Whether people have a right or not is sort of irrelevant if you see government as having a duty to materially improve people's lives.
A lot of people think that specifically is not the government's duty, though. You'd have to first convince them that the government's duty isn't simply to defend against invasion, or enforce the will of the people, or whatever else they believe.
While I would love to agree with you, the way I read the 14th amendment is that the government can not do anything (or pass any laws) that would deprive you of those rights. It does not imply (in my opinion) that they are required to do things to ensure you have those rights.