This reminds me of a similar situation in 1974 (I think). The government of Israel offered to fund the Yeshivas (education is something else the community has a chiyuv to provide). Rav Schach was instrumental in convincing the Yeshivas not to accept the government’s proposal. He felt, based on an opinion of the Chofetz Chaim, that when the government takes over the responsibility for funding, people get out of the habit of supporting those institutions that need our support.
Instead of asking the government to fund health care, wouldn’t it be nicer if we could take the tax money we would save and give it to Jewish social service agencies that help provide health care for poor people?