WASHINGTON — House Republicans emerged from a closed-door meeting on Friday with no new strategy to end the budget standoff and an angry plea to President Obama to negotiate over his health care law.

“This isn’t some damned game,” said Speaker John A. Boehner, his voice rising in anger. “The American people don’t want their government shut down, and neither do I. All we’re asking for is to sit down and have a discussion, reopen the government and bring fairness to the American people under Obamacare.”

Four days into a crisis that has shuttered much of the federal government — and 13 days before the nation faces an even more serious deadline to raise the statutory borrowing limit or risk defaulting on its debts — Congress appeared no closer to a resolution. Mr. Boehner opened the meeting of his fractious conference by reading letters from students at a Catholic school in Washington about how they handle stressful situations.

“We are locked in an epic battle,” he said.

The House on Friday will continue to plow through a series of minibills to reopen parts of the government, passing measures to finance the Federal Emergency Management Agency, the National Weather Service and nutrition services for women, infants and children. In a rare Saturday session, the House will vote on a bill to assure federal workers that they will receive back pay once the government is reopened. Next week, the House will consider a bill to finance Head Start, typically a target of conservative budget cutters.