The duty to die thing seems to have come from a speech that then Gov. Richard Lamm of Colorado gave way back in 1984. Lamn did use the phrase "duty to die", but he was talking in the context of people making the choice to end their own lives at some point not the government making the decision for them.
And it is a topic for discussion in bio-ethics circles. It is a rather obvious question after all. How far does your right to health care extend? How much money should the government spend for health care? Is it worthwhile spending limited medical resources extending the life of those with terminal conditions? If you don't believe your right to medical care is unlimited, then at some point you have a duty to die.
For the radical right it is just a rhetorical point however. These same people who argue the left is for government mandated death panels are for capital punishment, less regulation of insurance and health care, and would like to prevent illegal immigrants from getting health care at all. They are just selectively and intentionally misunderstanding the other side to produce a sound bite.
Jay