The GOP can't possibly win this stalemate, can they?
The government is shutdown but the ACA moves forward. If it stopped or slowed the ACA, that would be one thing. But the only thing this can do is greatly damage their party. Eventually, maybe only within a few days, the public will tire of this ploy which has nothing but a negative effect.
Per the Sean Hannity show, the latest plan by the House Republicans is to pass individual spending bills- the way they passed the one for the military last night, and then dare the Dems in the Senate to vote against. For example, they might pass a bill today which authorizes immediate spending for the national parks which would allow them to open again. If the Senate refuses, then the House can point at the Senate and say, "THEY'RE the ones who want to punish us," etc.
Hannity loves this; he's egging them on. Personally, I think this is all theater, and that Boehner is looking for a way to gracefully surrender and at the same time not lose his job. Not sure this is possible...