It's an interesting question, but I think the answer will ultimately depend on the details and fine print of the legislation that makes this happen.
'Universal health care', by itself, only means that everyone would have access to health care - it's not really a system for healthcare, but rather a goal of it. This goal could conceivably be achieved using our current system, i.e. by subsidizing the existing system in a way that allowed everyone to have access. In this scenario, we likely wouldn't see a huge change in the medical coding field. But if the goal is going to be achieved through more radical changes, such as moving to a single payer system, or a move to a nationalization of the healthcare industry along the lines of what's done in some other countries, this would naturally also involve major changes to the coding field, just as it would to all other sectors of the industry.
So it's hard to say what the future of coding until we actually see what those changes are in writing. In my opinion, however, due to the political climate and the pressures than can be applied to the legal system by groups that have a vested interest in preserving the existing system, major changes that disrupt our current system substantially have a very low probably of success and aren't something that we're likely to see any time soon, if ever.