On the other hand, the U.S. Chamber of Commerce has urged Republicans to fund the government and prevent a default, then double back and try and work out changes to the health care law later.
In a tea party age, it is unclear how much political clout establishment groups carry with individual GOP lawmakers.
At the same time, many Republicans fear a replay of twin government shutdowns nearly two decades ago that inflicted significant damage on their party and helped resurrect then-President Bill Clinton's political fortunes.
"When it comes to shutting the government down, that is not going to succeed because people don't want a government shutdown. And they'll blame Congress. They did before," said Sen. John McCain, R-Ariz.
Another Republican, Sen. John Cornyn of Texas, agreed. "I'm one who doesn't believe that a shutdown does anything except divert attention from a president and his policies, which are rightfully unpopular, to congressional incompetence," he said.
Cruz saw it differently hours earlier.
He told the National Automobile Dealers Association that the 1995-96 episode was just a "partial, temporary government shutdown" that didn't hurt Republicans politically as much as most people think and that it helped bring welfare reform in 1996 and a budget deal in 1997.
"Nobody likes that outcome. But it also wasn't the end of the world," he said of a possible shutdown.
Ironically, it was a Cruz comment that ignited anger among House conservatives on Wednesday evening.
As word spread that the House would pass legislation to fund the government and cut off money for the health care law, he issued a statement that said Senate Majority Leader Harry Reid "likely has the votes" to delete the health care provision. "At that point, House Republicans must stand firm, hold their ground, and continue to listen to the American people," he said.
House Republican aides said rank-and-file lawmakers on the House floor at the time vented their anger at what appeared to be a pre-emptive surrender.
Rep. Sean Duffy, R-Wis., tweeted that Lee and Cruz "refuse to fight. Wave white flag and surrender."
Rep. Lynn Westmoreland, R-Ga., tweeted, "Senate R's already declare defeat... before the battle even begins. So much for standing up for the American people."
Associated Press writers Ken Thomas, Donna Cassata, Alan Fram and Andy Taylor contributed.