My nuanced answer has been that if it were a "right," there would be a lot of poor countries out there violating its citizens rights for economic reasons alone. A right is not an entitlement but a freedom from a potential restriction.
This.
Health care and other social resources such as water and sanitation fall into an odd category of "Positive", or "Secondary" rights, rights which are essentially pre-requisites to classical "negative" rights. (Positive rights dictate a type of entitlement, whereas negative rights entitle the holder to lack of interference, which society is obligated to honor). Governments, then, have an obligation to uphold these secondary "rights"
to the best of their abilities as a means of upholding the "primary" civil rights of their citizens. To provide a working example: If a lack of health care or water restricted the activity of citizens, which in turn precluded their civic participation, and an active decision was made not to provide either resource, that action could by extension be seen as an infringement of a negative, or conventional right to civic participation.
This principle is borne out in the constitution of South Africa, which details the responsibility of the government to provide health care, water, etc to citizens. Can the South African government entirely fulfill its obligation? Of course not, it often falls laughably short. Does this mean that the government is perpetually responsible for violating the rights of its citizens? No. By making a good-faith effort to deliver on their obligation through responsible policy decisions (rationing, cost-cutting, etc.) it's able to uphold the rights of its citizens by providing a reasonable level of access to resources.
In short, health care can be thought of as a right, but mainly in the sense of an ancillary nuance of a social contract.
There are certain circumstances, though, in which I think health care can take the form of a conventional (negative) right. If the right to health care can be thought of as an extension of a right to physical integrity, then I think insurance rescission and patient dumping can both be thought of as blatant violations of the right to health care. Each involves a level of active interference that negative rights prohibit.