"If we get in the habit where a few folks, an extremist wing of one party, whether it's Democrat or Republican, are allowed to extort concessions based on a threat [to] undermine the full faith and credit of the United States, then any president who comes after me, not just me, will find themselves unable to govern effectively," he said.
The White House said Obama will truncate a long-planned trip to Asia, calling off the final two stops in Malaysia and the Philippines.
The shutdown also intruded into the race for governor of Virginia.
Terry McAuliffe, the Democrat, said he supports legislation to guarantee retroactive pay for furloughed federal employees. The Republican contender, Ken Cuccinelli, called on members of Congress to decline their pay as long as the shutdown lasts.
Obama's call to lawmakers to meet drew a quizzical response from Senate Republican leader Mitch McConnell. A spokesman, Don Stewart, said, "we're a little confused as to the purpose."
Boehner was "pleased the president finally recognized that his refusal to negotiate is indefensible," said his spokesman, Brendan Buck. "It's unclear why he'd be having this meeting if it's not meant to be a start to serious talks between the two parties."
The House sidetracked legislation Tuesday night to reopen some veterans programs, the national parks and a portion of the Washington, D.C., municipal government. All three bills fell short of the two-thirds majority needed because Democrats voted overwhelmingly against them.
Republicans tried again, this time under rules requiring only a simple majority. The parks measure was approved on a vote of 252-173, with 23 Democrats breaking ranks and voting in favor. The vote to reopen the NIH was 254-171. The House also voted to allow the District of Columbia government to use the taxes it collects to operate programs.
Votes were deferred on more bills, one to assure pay for members of the National Guard and Reserve, and another to allow some veterans programs to resume.
The NIH bill was added to the day's agenda after Democrats had said seriously ill patients would be turned away from the facility's hospital of last resort, and no new enrollment permitted in experimental treatments.
Democratic Rep. Louise Slaughter of New York said the Republican response was a ploy. "Every time they see a bad headline, they're going to bring a bill to the floor and make it go away," she said.
Some Republicans took obvious pleasure in the rough rollout Tuesday of new health insurance markets created under Obama's health-care law. Widespread online glitches prevented many people from signing up for coverage that begins in January.
Rep. Trey Radel of Florida said a 14-year-old could build a better website "in an afternoon in his basement."
Not all Republicans felt the same.
Rep. Peter King, R-N.Y., accused tea party-backed lawmakers of trying to "hijack the party" and said he senses that a growing number of House Republicans -- perhaps as many as 100 -- are tired of the shutdown that began Tuesday morning and will be meeting to look for a way out.
An earlier attempt by Republican dissidents to take control of the floor and vote alongside Democrats to reopen the government fizzled badly earlier in the week, and it was unclear if a new attempt could gain traction.
At issue is the need to pass a temporary funding bill to keep the government open since the start of the new budget year on Tuesday.
Congress has passed more than 100 temporary funding bills since the last shutdown in 1996, almost all of them without controversy. The streak was broken because conservative Republicans have held up the current measure in the hope of derailing or delaying Obamacare, just as the health insurance markets at the heart of the law opened on Tuesday.