Most politicians, and probably most Americans, see health care as a right. Thus, whether a person has the means to pay for medical services or not, he is nonetheless entitled to them. Let's ask ourselves a few questions about this vision.
Say a person, let's call him Harry, suffers from diabetes and he has no means to pay a laboratory for blood work, a doctor for treatment and a pharmacy for medication. Does Harry have a right to XYZ lab's and Dr. Jones' services and a prescription from a pharmacist? And, if those services are not provided without charge, should Harry be able to call for criminal sanctions against those persons for violating his rights to health care?
You say, "Williams, that would come very close to slavery if one person had the right to force someone to serve him without pay." You're right. Suppose instead of Harry being able to force a lab, doctor and pharmacy to provide services without pay, Congress uses its taxing power to take a couple of hundred dollars out of the paycheck of some American to give to Harry so that he could pay the lab, doctor and pharmacist. Would there be any difference in principle, namely forcibly using one person to serve the purposes of another? There would be one important strategic difference, that of concealment. Most Americans, I would hope, would be offended by the notion of directly and visibly forcing one person to serve the purposes of another. Congress' use of the tax system to invisibly accomplish the same end is more palatable to the average American.
True rights, such as those in our Constitution, or those considered to be natural or human rights, exist simultaneously among people. That means exercise of a right by one person does not diminish those held by another. In other words, my rights to speech or travel impose no obligations on another except those of non-interference. If we apply ideas behind rights to health care to my rights to speech or travel, my free speech rights would require government-imposed obligations on others to provide me with an auditorium, television studio or radio station. My right to travel freely would require government-imposed obligations on others to provide me with airfare and hotel accommodations.