Yes, the individual mandate is, historically, a Republican idea. Remember that the problem it seeks to address is access to health insurance. A single payor system addresses the access issue, but Republicans have historically sought to do so by means of market reforms. Key to any such reforms is the elimination of the "free rider" problem - where folks choose to go without insurance and are then hit by emergency or catastrophic care needs that hospitals are legally required to provide without compensation. Those uncompensated costs get passed on to others - primarily, the rest of us who act responsibly and obtain health insurance.
The problem with ObamaCare isn't the individual mandate per se, but its insistence that every American obtain soup to nuts health coverage that covers far more than emergency and catastrophic care. ObamaCare doesn't just require you to have insurance to cover costs that hospitals are otherwise required to provide on an uncompensated basis. It requires you to purchase health insurance to cover free vaccinations, routine doctor's visits, prescription drugs, chiropractic care, and a host of other things that many would rationally choose to self-finance. (Just as no one rationally chooses to buy auto insurance that covers oil changes.)
The individual mandate, so expanded by ObamaCare as to be the equivalent of a government mandate to "eat your broccoli", is likely unConstitutional. Which is unfortunate, since the SCOTUS is unlikely to base its ruling on the limiting principal noted above. And if the individual mandate is gone, then there will be little practical means to addressing the access issue other than to move to a single payor system, likely financed by general tax revenue rather than insurance premiums.