+Caring for the sick and infirm is a Christian's concern.
Considering the situation in the USA, I think when a government lies to its people about health care, Christians should evaluate their participation in both the government and its plans.
"Should a Christian be involved in health insurance?" is a question few even bother to consider. Is government directed health insurance what God would want us to do in answering the call to care for each other? Can we look to the government as fulfilling our responsibilities when we know the government lies with impugnity?
Perhaps it is a good time in the USA to reconsider all of this and look at alternatives like the Amish method of health care, the old Lodge doctors once known in the USA, or Christian MediShare which has been growing quickly since its method is Biblical and it is an "out" written into the Obamacare law... for now.
Even if you live in the States and have adequate health coverage, do look at what Christian Care provider MediShare is doing. Two similar ministries doing roughly the same thing and also exempt through the Obamacare law are Samaritan Ministries and Christian Health Ministries and both deserve consideration as a model for how this is done or as an option my American readers may wish to pursue.