Perhaps therein lies the problem. The only way to absolutely fulfill what they see as a universal "right" (as opposed to privilege) is to have Gov't step in.
The Pope says it is the moral responsibility of nations as opposed to the charity of individuals which implies that the State should step in.
The Vatican Sec'y of State (I assume he's a spokesperson for the Church's position) seems to imply so in this same story
the Secy of State says "Justice requires guaranteed universal access to health care,"
Then goes on to say:
Governments are obligated, therefore, to adopt the proper legislative, administrative and financial measures to provide such care along with other basic conditions that promote good health, such as food security, water and housing.
As Pius XI said:
"We make this pronouncement: Whether considered as a doctrine, or an historical fact, or a movement, Socialism, if it remains truly Socialism, even after it has yielded to truth and justice on the points which we have mentioned, cannot be reconciled with the teachings of the Catholic Church because its concept of society itself is utterly foreign to Christian truth.
(Encyclical Quadragesimo Anno, May 15, 1931, n. 117)
[Socialism] is based nevertheless on a theory of human society peculiar to itself and irreconcilable with true Christianity. Religious socialism, Christian socialism, are contradictory terms; no one can be at the same time a good Catholic and a true socialist. (Ibid. n. 120)