How would government shutdown affect Obamacare?

The Washington Examiner's Tim Carney warns that most Republicans seem to be confused about the effect a potential government shutdown would have on the implementation and enforcement of President Obama's unpopular health care law. On the one hand, some Republicans worry that forcing a government shutdown over Obamacare will ultimately hurt the party, while some others think taking a strong stance against Obamacare may actually help.

So who's right?

Carney argues:

Shutting down the government won’t defund Obamacare: Almost all Obamacare spending is “mandatory spending.” Obamacare authorized that spending until the end of time, and it doesn’t need to be appropriated every year. So under a government shutdown, we’d run out of discretionary spending, but most of Obamacare would keep humming.

A continuing resolution could defund Obamacare: Just because it’s “mandatory” doesn’t mean it’s invincible. A continuing resolution could contain a “policy rider,” as appropriations bills often do, which really does defund Obamacare.

Defunding Obamacare would require cooperation of the Senate and the White House: Because you’d need to pass a CR defunding Obamacare to actually defund the mandatory-spending portions of Obamacare, that would need to pass both chambers and get the White House signature. You can’t defund Obamacare without Obama.

Conservatives say they think Obama can be coerced into defunding Obamacare. Will Obama let his whole government go defunded just to save this one unpopular law?

Given the fact that the majority of Americans want to avoid a government shutdown, I'm inclined to agree with Carney: Obama would likely let the government shut down before defunding Obamacare.