The abstract of a paper (ungated verson here) by Jonathan Gruber and Kosali Simon in the Journal of Health Economics:

Ten years ago, Cutler and Gruber [Cutler, D., Gruber, J., 1996. Does public health insurance crowdout private insurance? Quarterly Journal of Economics 111, 391–430] suggested that crowd-out might be quite large, but much subsequent research has questioned this conclusion. Our results using improved data and methods clearly show that crowd-out is still significant in the 1996–2002 period. This finding emerges most strongly when we consider family level measures of public insurance eligibility. We also find that recent anti-crowd-out provisions in public expansions may have had the opposite effect, lowering take-up by the uninsured faster than they lower crowd-out of private insurance.

Something to keep in mind when candidates prattle on about universal coverage--what they really mean is universal government coverage.