My number 1 objection is having a government database with my private information, the assumption that it will never be misused or abused in any way is preposterous.
My number 2 objection is the reality that the US is down some on mortality of infants and in some other categories because we are often willing to push for risky strategies to attempt to save people that may not have had that chance in other places. Such risks tend to be avoided by government programs, shoot some of them consistently speak of lower technology because of the programs (see France).
We have a very high infant mortality rate in the US for a nation that is so advanced with our medical technologies. In fact our advanced status is precisely why we have a high infant mortality rate. Statistically if we did not attempt life saving measures on preemies they would have died and been considered miscarriages not infant deaths.
I agree with you that these kinds of efforts, specialized hospitals etc. would be some of the first kinds of medical expenses to disapear from a government program.