Some poorer women in the US don't bother getting pre-natal care, even though it's available through Medicaid. So that government health care that already exists isn't helping them in any way. As for higher rate of patient deaths, that's probably because there are more risky procedures done in the US, because there's more innovation with the private health care facilities.
One way to judge the two systems is to see how many US doctors are moving to Canada to practice medicine (I've never heard of any), compared with Canadian doctors heading south to the US.
More physicians returning to Canada The number of physicians who moved abroad decreased by 57% over the past five years. Furthermore, for the third year in a row, the number of physicians returning from abroad in 2006 was greater than the number leaving Canada (238 compared to 207). CIHI’s study also found that most of the overall increase in physicians is attributed to an increase in Canadian-trained physicians. The number of doctors trained in Canada grew by 5.2% between 2002 and 2006, compared to an increase of less than 1% (0.7%) in the number of foreign-trained physicians. Doctors trained abroad come primarily from the United Kingdom and Ireland, South Africa, India, Egypt and the United States.
http://secure.cihi.ca/cihiweb/dispPage.jsp?cw_page=media_25oct2007_e