Boehner has some capital. Obama was furious when the Speaker walked away from the table last time. And he would be perfectly happy do it again if Obama acts the same way.
Boner can go and play a game for the cameras. But I think it is risky because the Obama can lie and say Boner said this or Boner said that. What Boner really needs to do is sit back and watch the show. He will get some angry phone calls but not nearly as many as Feinstein, Boxer, and a bunch of other blue state senators. Sequestration is our major lever against the blue states.