It is really hard to say at this point whether House Republicans will do anything about immigration reform. One can speculate about their motivations, but it's interesting that the most damning account comes not from their opponents but from the Republicans themselves. Republicans now say they are going to refuse to do anything on immigration reform, not because they oppose it, but because they're feeling spiteful toward President Obama:
Other prominent immigration supporters like Sen. Marco Rubio (R-Fla.) have also backed off any deal, saying the Obama administration has “undermined” negotiations by not defunding his signature health care law. Rep. Raul Labrador (R-Idaho) went further, saying Obama is trying to “destroy the Republican Party” and that GOP leaders would be “crazy” to enter into talks with Obama. ...
“After Obama poisoned the well in the fiscal showdown and [House Minority Leader Nancy] Pelosi now is actively trying to use immigration as a political weapon, the chances for substantive reforms, unfortunately, seem all but gone,” said one GOP operative involved in the conservative pro-immigration movement.
Let's unpack the argument here, because it's pretty amazing. It would be one thing if these sources opposed immigration reform on its merits. But they don't. They think immigration reform is a vital necessity. They believe failing to pass some kind of reform will harm America. But they want to do it anyway because they're mad about the shutdown.
This leads to a second, equally crazy view that Obama is to blame for the shutdown, and that Republicans are justified in their anger at his refusal to give Republicans concessions to reopen the government or avoid a default.
But that secondary belief is really a side issue. Suppose we assume Republicans are correct here, and Obama behaved outrageously by not submitting to policy extortion. The Republican argument is that this dastardly behavior justifies Republicans doing something they believe harms America. This is not an accusation against the Republicans. It's their own explanation!