If rights mean anything at all, there can be no "reasonable exception" to equal rights.
I think I agree on the concept of natural rights. What I'm getting at more is the "right to be treated equally" which seems a lot more tenuous. Usually when someone says they are for equal rights, they aren't just talking about natural rights. For example, being offered a job isn't a natural right. However, if someone started discriminating when hiring based on sex or race or other protected criteria, we would treat that as a rights violation. But there are cases when discriminating makes sense, like not letting a blind person drive a car. So these aren't reasonable exceptions to natural rights, but reasonable exceptions to protections that the government provides on top of natural rights.
Rights aren't determined by accountants
I'm also not sure universal healthcare is a natural right, it depends on what the system actually achieves and if it actually improves health outcomes. It can be a good thing if designed well, but the financial costs are not irrelevant since people are taxed to fund it.