To close approximation, the minority party in the House occupies legislative Siberia, but there are a few ways it can make its presence recognized. One is by filing what's known as a discharge petition on a piece of legislation. Get 218 signatures and that bill gets a vote on the floor, whether the majority leadership likes it or not.
The theory is that if the House fails to pass an immigration bill of its own and then John Boehner refuses to put the Senate bill on the floor, 17 rogue Republicans will rebel (or get quiet dispensation) and agree to sign a discharge petition. Voila.
There are a lot of reasons why I think this is unlikely. In a partisan and polarized House discharge petitions in general aren't much more than clever messaging documents. Signing on is an affront to the majority leadership, and In the case of immigration Republicans who signed on would face external consequences, even if leadership tacitly give it the go ahead.
But even if a Dem discharge petition is marginally more likely to be effective on immigration than on other issues (and I'm not saying it is) GOP leaders would still have a huge incentive to do everything in their power to keep their members from signing on.
To be a bit crude about it, the only reason for the Republican party to greenlight a bill that will make 11 million current immigrants citizen is if they get to share in the credit. Killing it altogether would have all kinds of consequences, but the two upshots for Republicans would be that a). those immigrants won't become citizens anytime soon, and b). they could at least attempt, however disingenuously, to convince the public that the failure was bipartisan.
Those silver linings disappear if Democrats manage to get the bill enacted "by accident."
In fact, the worst possible outcome for the GOP would be for the bill to become law over the explicit objections of leadership. It would give Democrats a huge policy victory but leave Republicans without the political dividends they'd pocket by being equal partners in the reform effort. It might even exacerbate their problems with Hispanic voters. And allowing a couple dozen Republicans to sign a discharge petition would accomplish just that.