I learned long ago that in politics, while it is tempting to vanquish your enemies, sometimes a little twig from an olive branch can work wonders, and provide a face saving way for your foes to retreat, so you can both then move forward.
Obama could agree to postpone the penalty on individuals for non enrollment for one year, to work out some of the kinks that may be coming apparent in implementation of the law. He already granted corporations a one year hiatus for the same reason. He should do this not because republicans have shut down the government, but because it is the right thing to do, and couple that with a vote on the debt ceiling limit, and call it a day. Hold Boehner to his pledge to avoid default, and kill two birds with one stone, and then move on to solving other problems, like immigration and infrastructure development.