@oristarA,
Well, private health insurance would be all that is not operated by the federal and state governments. That would include employer provided insurance, and unemployment is fairly high at the moment. Not all employers have health insurance benefits, but the decline in employment could account for most of the decline in the numbers of privately insured.
I seem to recall that some 46 million are without insurance at all, including both government and private insurance. As you've implied, the principle of adverse selection applies. Those who are nominally too healthy to need insurance as a group are unwilling to pay for it. Those who are at high risk often cannot pay for it, if the can get insurance at all.