This is a very childish approach for a couple reasons.
First, it's easy to say let unemployment rise to teach the Dems a lesson. Politically good, but for all those many thousands of people/families put out of work, those are individuals who will be suffering. Pink-slips are neither red nor blue, so some Obama voters will get hurt and some who voted against him will get hurt.
But, just keep in mind, it was the spineless Boehner, who caved in 2011. He could have chosen to take the political hit then, and called the Dem's bluff to shut down government. Instead, he chose to go along for the scenic mountain drive leading to the inevitable fiscal cliff.
Don't cry for me Boehner, and don't try to pretend you've miraculously grown a spine!