First, the Republicans spend months planning the crisis, so much so that it was an open secret and everyone knew that it was coming. They even told everyone the game plan (or lack of plan). Then, once in the furlough, proceeded to blame the Democrats.
The fault really does lie with the Republicans. They set out a game plan which was this: if they defund Obamacare (the ACA), they win. Otherwise, they lose. That's was it. Given that really bad set of rules, its no wonder they fought tooth and nail. In baseball, that's like saying, "Home run in the 9th, we win, all other results we lose, even if we scored more runs." It's crazy stuff.