Too much could be written about this argument but to me a right can't be provided by someone else's labor.
Now that I think about it, a right is also something that can't be taken away. No one should be able to prevent your free speech (1st Amendment), defend yourself (2nd Amendment), or violate your privacy (4th Amendment) without due process.
In this case no one is preventing someone from being able to care for themselves. We are just arguing over who pays for it. It is probably a bad example but everyone has the right to use the roads for transportation. We just don't have the right to the car, insurance, and gas.
Edit: Added more comments