There isn't a single government agency or division that runs efficiently; do we really want an organization that developed the U.S. Tax Code handling something as complex as health care?
"Free" health care isn't really free since we must pay for it with taxes; expenses for health care would have to be paid for with higher taxes or spending cuts in other areas such as defense, education, etc.
Profit motives, competition, and individual ingenuity have always led to greater cost control and effectiveness.
Government-controlled health care would lead to a decrease in patient flexibility.
Patients aren't likely to curb their drug costs and doctor visits if health care is free; thus, total costs will be several times what they are now.
Just because Americans are uninsured doesn't mean they can't receive health care; nonprofits and government-run hospitals provide services to those who don't have insurance, and it is illegal to refuse emergency medical service because of a lack of insurance.
Government-mandated procedures will likely reduce doctor flexibility and lead to poor patient care.
Healthy people who take care of themselves will have to pay for the burden of those who smoke, are obese, etc.
A long, painful transition will have to take place involving lost insurance industry jobs, business closures, and new patient record creation.
Loss of private practice options and possible reduced pay may dissuade many would-be doctors from pursuing the profession.
Malpractice lawsuit costs, which are already sky-high, could further explode since universal care may expose the government to legal liability, and the possibility to sue someone with deep pockets usually invites more lawsuits.
Government is more likely to pass additional restrictions or increase taxes on smoking, fast food, etc., leading to a further loss of personal freedoms.
Patient confidentiality is likely to be compromised since centralized health information will likely be maintained by the government.
Health care equipment, drugs, and services may end up being rationed by the government. In other words, politics, lifestyle of patients, and philosophical differences of those in power, could determine who gets what.
Like social security, any government benefit eventually is taken as a "right" by the public, meaning that it's politically near impossible to remove or curtail it later on when costs get out of control.