I think there is an error that many make when speaking on social issues and health care is no different. The Bible speaks of caring for widows and orphans and others in a number of places; Deut 10:18; Deut 14:28-29; Deut 24:17-21; Ps 82:3; 1 Tim 5; James 1:27; etc. It is often verses such as these that those purporting a social gospel grab hold of to say the government needs to do something. But this misses the point that these actions were put forth to the church and Israel and not set up for the government of the land. Yes the government of Israel would have part in this as in Deut 14:28-29 but we could probably see this as even now applying to the church but not to the government as we have today.
Another thing that is also often missed is that the care that is to be offered often has stipulations as with widows in 1 Tim 5 and with those being able to work in 2 Thes 3:10. Even the offering in Deut 14:28-29 is only every three years so is not perpetual in the sense of most programs today. To simply have a social program or health care open to all with no stipulations is not biblical.
The bottom line, scripture does call for believers and the church to help those truly in need as the bible directs and describes but simply because we are called to do so does not dictate the government regulating it. The church and those that like to use the church have misused God’s call for his people and in many ways all people to mandate a top down manner of caring for others. Care for others is to be done as an out flow of the commands of God and His change of our hearts. This is the call in Luke 10:30-37 but that call is not for a program to help those in need but is to be ready to care for others as the need arises as the Samaritan did. To use this passage, as I have seen done, to say universal health care is a biblical principle misses the point Christ is making, that we are to all care for our neighbor out of love for God not out of subservience to the state.
So I would say that to say the church needs to be involved in some manner with helping the needy is biblical but it needs to be done out of obedience to scripture not some governmental law. I would even say that this is truly how things have worked in the past, that is until we as a people began to rely on the state to take care of these issue. It was the church that set up orphanages and feed the hungry. But then the state started taxing so there was less money and they took over the welfare and the church handed that over to the state so look where we are now.
As an addition:
While I would say that National Health Care would be in violation of the 10th amendment of our constitution we as believers need to first seek what the Bible says before we consult even the constitution. However in many instances you will find the constitution when properly interpreted through the lens of its framers will in general not conflict with the scriptural mandate. However interpreting the constitution through the “living lenses” of our current government is another matter and is very often in conflict with the Bible.
So at the end of the day while health care may be good it is not a right and God has set up His kingdom to be cared for by His people in His way. As with the Samaritan we are to care for the needy as God brings them into our midst but we are to do so as a people of God and the church of God, His bride, not as a mandate of the state. Once the conversation strays from the Bible it becomes mostly opinion but when we stick to the only infallible and unchanging standard then we can make God glorifying decisions.