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Following a two-hour closed-door GOP meeting on Tuesday, Rep. Mick Mulvaney, South Carolina Republican, marveled at “how calm it was and friendly” in the meeting, and at the way Mr. Boehner has juggled the various factors of the GOP caucus.

“It was a conversation that we could not have had six weeks ago,” Mr. Mulvaney said. Asked why, he said that “there was not much cohesion in the conference,” but that Mr. Boehner helped change the tone by making sure that everyone had a voice.

“I think you get to the point that you feel like you can participate and when you have a meeting like that that, your leadership will listen to you and take it into consideration and that has really helped us come together as a group,” Mr. Mulvaney said.

When Mr. Boehner emerged with a plan that would make several dents in Obamacare, including repealing a despised tax and requiring President Obama and Vice President Joseph R. Biden to sign up for the health exchanges, Democrats were furious.

Mr. Reid accused Mr. Boehner of putting his own political future ahead of the country and of trying to scuttle negotiations Mr. Reid was having with his GOP counterpart, Sen. Mitch McConnell.

“I am very disappointed in John Boehner, who once again tried to preserve his role at the expense of the country,” Mr. Reid said

Underlying much of the debate about Mr. Boehner is how he handled Sen. Ted Cruz, the Texas Republican who, along with some vocal allies, wanted to reject funding the government until the president canceled Obamacare.

Mr. Boehner early on announced he wouldn’t go that route, and even struck a deal with Mr. Reid. But he was force to recant after enough Republicans insisted he try.

“In all fairness, this wasn’t John Boehner’s strategy,” Mr. McKenna said. “It was a battle he didn’t really want on a battlefield he didn’t pick, and he did what he could with it. Every honest member of the caucus would say that’s right.”