As a Republican, you have to be worried about this. I am surprised to see them challenging Obama in a way that mirrors the negotiations with Clinton 16 years ago so closely. Their position isn't even as strong now as it was then: at that time, they controlled both houses of Congress. Moreover, they did exactly what I would have recommended, which is to pass a budget and dare the president to veto it. Then, I thought, the resulting government shutdown would clearly be Clinton's fault, and he would get the blame. Instead, Clinton simply said that the Republicans needed to send him a bill "that I can sign," and for some reason the Republicans ended up taking the bulk of the blame (although they did retain Congress in 1996).
Now, the Republicans can't even send Obama a bill because the Senate won't pass the budget approved by the House. Moreover, the Republicans seem even more likely to get the blame this time, because they are absolutely refusing to negotiate on a key point, which is raising taxes. If a deal fails, it seems they are set up to take responsibility.
Of course, things rarely repeat themselves that closely. This scenario could happen, but something quite different could also occur. I wonder, for example, if Obama's failure to present his own budget will be a serious liability in the event of failure. Then, too, spending has unquestionably gone through the roof during his administration, even not counting the stimulus. It would take massive spending cuts just to bring the budget back to where it was in 2008, so it's not as though Republicans are proposing gutting the government as it has long been.
Still, I think any Republican would have to be nervous about how the debt ceiling negotiations will play out. I know I am.