Native Americans generally decide who is and who isn't apart of their tribes, usually based on blood and having a connection to the tribe. They're considered sovereign and generally take care of their own needs like healthcare.
The 14th Amendment says all person's born on US soil are citizens. We've had several Supreme Court cases over whether blacks, Asians, etc were really meant to ever become citizens. So if we ever get nationalized healthcare, all US citizens would be eligible for it.