While the House can pass a budget with no funds for Obamacare, it will face the same fate in the Senate, and the media will have a field day blaming republicans as the deadline for "government shutdown" approaches.
I suppose its worth a try, but I cannot see the Senate yielding to pressure when the forces at large are all arrayed against the House republicans. Reid would call Boehner's bluff and Boehner would fold because his heart isn't really in this anyway.
No one has proposed shutting down the government
Cruz and Lee have both said they will vote for a CR (whatever happened to budgets?) without ObamaCare funding
It is a shame our strategy has to be based around Boehner's being a gutless coward.