The only way to prevent a pro-illegal, anti-border security bill from being passed into law is to not pass any immigration bill out of the House at this time.
Any bill passed by the House will be gutted and rewritten when it goes to the conference committee for resolution. It will end up with guaranteed amnesty and no mandatory closing of the border.
Once a bill is reported out of conference and sent back to both houses for a vote it becomes problematic for the senate or house to vote it down, but more so for the republican house.
Especially in the today’s political environment where the major media is pushing for pro-illegal legislation and basically acting as the public relations department of the democrat party.
Unfortunately, if the House does not pass something, there is another way for RINOs. They can vote with House Dems to pass the senate bill. This has happened recently on the VAWA.
WTH??? "Suggested?" "majority of both parties’ support?" I suspect that is going to boil down to Dems+RINOs, but it might mean that each party would have to have a majority supporting it. Maybe that's the best Boehner can communicate, but maybe he doesn't want to be clear.
NOTE I don't have a working crystal ball. Sometimes fate changes for reasons none of us predicted. For example, in the 2007 amnesty McCain could have killed a GOP poison pill that defeated Reid but he was absent, gone to a fundraiser for his POTUS campaign.
All of us should watch GOP congresscritters carefully, and inform them that we are watching them, even if they are behind closed doors in some senate-house conference committee (the House part of which will probably have a bigger % of RINOs than the whole House membership). Some of them may not care, but I think that's the best shot we have.