I've heard a republican idea that would instead allow insurance companies to charge a large fee for lapses in coverage, which would be nearly equivalent to requiring coverage except people would be hit with a big fee when signing up.
Medicare already does this. If you go uninsured and then try to enroll (at a time when you're eligible, of course), there's a penalty you're required to pay.
I didn't say it was a good plan, just one I heard suggested. I'm in the opinion that the US should just combine to have one giant risk pool, and include everyone in it (aka single payer)