With regard to the idea of whether you have a right to health care, you have realize what that implies. It's not an abstraction. I’m a physician. That means you have a right to come to my house and conscript me. It means you believe in slavery.
Senator Paul makes one good point. A claim of a right to health care is a claim on someone else's property, skill, and labor. Indeed, a right to health care does imply the ability to "conscript" the service of health care providers. Though public conscription of private property and labor are permissible under our system of constitutional government---purchasing property through the power of eminent domain and a military draft spring to mind---doing so imposes substantial costs on individual freedom and should not be taken lightly or without the involvement of compelling public interests. The burden that a right to health care places on health care providers is, I think, rarely considered by its advocates. Senator Paul is entirely correct to draw attention to it.
Senator Paul is also completely wrong to equate the conscription of health care providers with slavery. A slave is the property of another person. A slave has no choice in the work he or she does and not control over the conditions under which he or she lives her life. Slaves have no more legal standing than livestock. The creation of a statutory claim on health care services would not make doctors, nurses, physicians assistants, or other health care providers slaves in any reasonable sense of the word. It is silly and politically tone deaf to say so.
Matt Welch expresses some similar thoughts at Reason's Hit & Run Blog.