The United States a a system of Government with a separation of powers. The executive wing does not need a majority in the legislative wing (like in NZ). This model has both strengths and weaknesses to it.
One aspect is that a budget needs to pass the House, the Senate and the President. Since the 1990s it has been hard to achieve this as if no party holds all three, they have different ideas on what the size of the state should be and appropriate levels of spending and taxation.
I’m all for the Republicans refusing to pass a budget that increases spending and taxation, especially as the USG has a deficit which is huge as a % of GDP. They absolutely need to have a realistic path back into surplus – which will at some stage mean entitlement reform. The across the board spending cuts that sequestration brought in have not had anywhere near the dire effect that Obama claimed. While they are a blunt tool, they have shown you can reduce spending (or even the growth in it) without a huge impact on services (as has been the case in NZ).
However I think they are on much weaker ground with their refusal to pass even a continuing resolution to roll over funding, unless Obama agrees to defund Obamacare. Like it or not he won that fight and got the law through a previous Congress. Refusing to pass a budget unless he abolishes it, or defers it, is unreasonable to me.
They won a good battle on the sequestration and that will increase the reductions in spending in the future. Obama hates the sequestration cuts and will be in a weaker negotiating position when the second round is about to hit. They may be able to get a deal at that point. But their demands for him to effectively abolish the signature law reform initiative of his first term is unreasonable to me. Their party is divided on it and may splinter. On the other hand, maybe Obama will blink and agree to a deferral as the law is still pretty unpopular.
Some Republicans think a US Government shutdown isn’t the end of the world, and they are right. While very embarrassing, and painful for affected employees, a couple of weeks or more of no museums and bureaucracy is not that big a deal. It has happened before.
What would be a big deal is a refusal to increase the debt ceiling, which would see the United States of America default on its sovereign debt. As they are the global reserve economy, this would be a massive deal. It could plunge the world into recession. The US Chamber of Commerce is against any games over the debt ceiling and I agree with them. The focus should be on cutting spending to reduce the deficit and eventually stop increasing the debt. But to refuse to increase the debt ceiling when you have already passed budgets that have deficits, is not reasonable.
So in summary the GOP should continue to hold firm on sequestration and not agree to end it unless they get a good deal on spending reductions. It is essential that spending be constrained as the US deficit is so massive. But they should not use the budget process as a way to effectively over-turn a law already passed. That sets a bad precedent for all future Congresses to do the same. And they also should not play games on the debt ceiling.