I've got no particular data to back me up, but since the majority of Americans already purchase health insurance, it seems reasonable to presume that the majority of Americans who oppose the "individual mandate" provision of the Affordable Care Act are people who already purchase health insurance (usually through their employer), and thus people for whom this mandate would have little or no impact.
And really, the mandate is the only actual provision of the ACA to which most Americans can voice any specific opposition. There are generalized concerns that the ACA doesn't do enough to reign in rising health care costs, or that it adds too much to the federal budget, or, you know: "Gov'mint!" But I don't see many people protesting in defense of, say, kicking young adults off their parents' policies, or for maintaining exclusions for pre-existing conditions.
The ACA is imperfect, no doubt. Republicans ignore this in touting their anti-ACA polls, but a good chunk of the negative opinion out there is due to the perception that the ACA didn't go far enough toward a public option or even single-payer system. But the reforms we got through are a huge step in the right direction toward greatly reducing the health care insecurity that plagues much of low- and middle-income America.
Yet somehow, an awful lot of voters have been snookered into believing that it would be better to throw themselves at the mercy of the nation's health insurance companies than to brook a "mandate" that they'd never willingly choose to opt out of. Again, curious.