Well we have actually defaulted before, back in spring of 1979 between two of the shutdowns. Something of an 'oops' moment more than brinksmanship but there is precedent. As to now, well I've no Crystal Ball. People are currently blaming the GOP more than the president as an average thing but they're both taking hits from the indies, obviously most GOP and Dem blame it entirely on the other side and they can bank on that support no matter how the wind blows. The indies though see this as a shads of gray argument with POTUS defending his unpopular bill and the GOP just being obstinate. In large measure the GOP's faring slightly worse in the public eye because Obama is seen as such a weak-kneed negotiator, especially post Syria/Russia that its assumed they are the ones not compromising since they think he rolls in a stiff breeze. As it drags on, I'd guess the GOP will be publicly saying what it wants and Obama will have to explain why he is standing his ground. If the GOP says "We want, in exchange for funding Obamacare, a removal of all the Exec Branch waivers granted from it" odds are good POTUS is screwed. The public is inclined to view those waivers as unfair and if he doesn't roll on them, in part, it makes Obamacare look even worse along with his position and makes it more like him desperately defending bad policy that has his name literally on it.

That's what I'd do anyway, demand all current and future waivers be subject to congressional approval within 90 days of it being granted by the exec and that congress can waiver anyone it wants by majority in both houses. It might not work and they might not do it. The government might shut down in 10 days, ask me in 11 and I'll tell you