It would seem to me that everyone has an equally just and lawful claim to health and well-being
This premise is false, and your argument falls because of it.
Consider the situation of a healthy young person. That person is in a state of health and well-being that requires nothing from anyone. It is the state of nature. Their "right" to their situation derives from the natural order.
Now consider a 75 year old man with acute myelogenous leukemia. In your formulation, his "just and lawful claim to health and well-being" is equal to the hypothetical young person.
But unlike the young person, this person's "just and lawful claim" is impossible of achievement, not by any expenditure, however great, nor by any human extertion, however forceful.
The "right to health care" (which does not exist, IMO) does follow from the Declaration on Human Rights, since the "right to health and well-being" does implicate the associated human exertions directed at health.
But the "right to health and well-being" in the DHR is directly imported from preexisting socialist systems, principally from the 1934 USSR Constitution, and this fictitious "right", granted to the well by nature and unavailable to the sick under any system, was invented precisely to extend State control over the medical sector.
Reflect on what can or should be done to secure the "right" for those with incurable illness, and you will, I think, advance your thinking on the subject.
You're right. I've called it the wrong thing. I wanted to refer to the right to health care, which is more appropriate to the discussion.
I do think we all have the right to health care. It has nothing to do with nature, but rather forms part of living in a modern society. As a society, as something separate from nature, we bestow these rights upon ourselves.