Cole said he voted against Obamacare and voted 40 times to repeal, reduce or delay it. But he does not want to shut the government down over the issue.
“I don't think it would work first of all,” Cole said. “It would also hurt a lot of good people.”
If the government is shut down, military members won't get paid and veterans' centers will close, Cole said.
Politically, Cole said he thinks the strategy would backfire, strengthening the president's position and turning voters in swing states against Republicans.
Cole said the House can chip away at Obamacare, but Republicans are going to have to win the issue at the election box.
“We're in the position to stop a lot of what he's doing, but we're not in the position to undo it until we have the patience and discipline to win multiple elections.”
Cole also heard from several frustrated Tinker employees who took furlough days because of the sequester budget cuts that took effect this year.
One man, who did not want to give his name, told Cole the federal employees being hurt by the cuts are tired of constant uncertainty about their jobs.
“I don't know what you are hearing from the leadership at Tinker, but morale out there is really low,” he said. “People are going to start leaving for other jobs.”
Cole blamed Obama for the situation. He said he is hopeful leaders from both sides can negotiate a solution that will prioritize the sequester cuts to keep Tinker employees from having to take furlough days next year.
“If we do get to a big deal, I guarantee you there will be stuff in it everyone doesn't like,” Cole said. “That's what it means to make a deal.”
Cole tried to explain to several upset questioners that the House can't unilaterally push its priorities into law when the Senate and presidency are controlled by Democrats. He also brushed off calls to impeach Obama and disagreed with a questioner who said the country had turned into a socialist state.
“California is 10 Oklahomas on the electoral map,” Cole said. “New York is five. Those people are as passionate about what they believe as the people in this room.”