Having a government-ran Healthcare system is a very socialist ideal.
I believe that having a government-ran Healthcare system would be very bad. We actually talked about this a little in my class, too. Essentially, yes, Healthcare is very costly and it is unfortunate that a lot of the people who need it can not afford it, though for these people, Healthcare can be easily obtained through a job (if they apply themselves and go work for what they need instead of eating out of garbage cans).
EDIT: Also, on the notion of government-ran institutions. The DMV is a state-governed agency and has nothing to do with federal government, and if it did it would be violating the United States Constitution. Also, these agencies suck fucking hardcore not because of the agency itself, but because the people are usually miserable at their piss-poor jobs.