Via Greg Sergant, it appears as if more founding fathers, including Thomas Jefferson, were supporters of socialized medicine, namely the "An Act for the Relief of Sick and Disabled Seaman" law signed by President John Adams in 1798.
Alexander Hamilton supported the establishment of Marine Hospitals in a 1792 Report, and it was a Federalist congress that passed the law in 1798. But Jefferson (Hamilton's strict constructionist nemesis) also supported federal marine hospitals, and along with his own Treasury Secretary, Albert Gallatin, took steps to improve them during his presidency. So I guess you could say it had bipartisan support.
Why did the founding fathers want a government takeover of healthcare? Death panels and the like.