http://www.usnews.com/news/blogs/god-and-country/2009/08/17/us-bishops-demand-universal-healthcare-without-abortion
The link for the Pope's position where he describes healthcare as an "inalienable right" is given in Post #1.
Calling something a “right” doesn’t equate to it being 100% paid by the federal government, though. You have the right to worship freely, but it doesn’t follow that the government is obligated to build a church for you to exercise that right.
It's the mistake of thinking that when the Church says "right," she means "State obligation." A basic mis-reading.
Just an example: It's like when the Church speaks of a "right to marry" and the gays dive in and say "Us too." Wrong-o. You have to define "marry" as the Church does (Sacramental union, man-woman, eligible to marry, e.g. not already married to somebody else, etc.) and "right" to mean "a natural good to which, in a good society, you have access."
If "right" in this context properly meant "the state must supply it," the State would have to supply spouses to all citizens, I suppose!
Catholics of a generation ago would see medical missions, Catholic hospitals, and fraternal insurance (e.g. Knights of Columbus) --- in other words, voluntary, mutual, and charitable inititives --- as responses to the right to health care as the Church uses the term.