I think you're making several very "odd" assertions here.
First of, "a person who doesn't want it" does not apply in this case with Terri since there were no written directives from Terri. She may well have wanted it. How do you know what she wanted at this point in time? You don't and no one else knew either.
Second, your post implies the two parties I am talking about -- parent and child -- are utter strangers, and therefor the entire claim to a right of care is ""odd." But your omission of the actual relationship is what is odd. A parent-child relationship is recognized by the court, and has spawned activists to band together to fight for parental rights in many situations, including education.
Third, as for your "Un-American" claim, here is what Supreme Court Judge Sandra Day O'Connor had to say about parental righs:
The above is from a web site called parentsrights.org
So, you see, "parental rights" is already well-known legal concept. It is not "odd" nor "un-American" nor between strangers, as you imply.
And, yes, I know Terri was married, and the courts recognized her husband as her legal guardian throughout this matter.
However, again, with no written directive from Terri, and with Terri's own parents willing to care for her, it seems to other parents that Terri's parents had a fundamental legal right that was overlooked here. And, without a written directive from Terri, the parents' right to care for Terri was superior to a spouse's right to pull the plug.