Jonathan Gruber, the Massachusetts Institute of Technology economist whose comments about the health-care law touched off a political furor, worked more closely than previously known with the White House and top federal officials to shape the law, previously unreleased emails show.

The emails, provided by the House Oversight Committee to The Wall Street Journal, cover messages Mr. Gruber sent from January 2009 through March 2010. Committee staffers said they worked with MIT to obtain the 20,000 pages of emails.

Speaker John Boehner (R-Ohio) is pushing back against the idea of Republicans simply continuing ObamaCare subsidies if the Supreme Court cripples the law.

At a press conference Thursday, Boehner was asked why a House GOP plan included repeal of the individual mandate, which would just be “veto-bait” for President Obama, and why Republicans would not just extend subsidies through the presidential election while looking for concessions elsewhere in exchange.

“Clearly, we’re interested in protecting those millions of Americans who could lose their subsidies. But, as I said, we are not interested in protecting a fundamentally broken law,” Boehner said.

“Protect the people, not the law,” said Sen. John Barrasso (R-Wyo.) in a brief phone interview, describing the mind-set of the GOP should the Supreme Court rule against the administration and strike down the federal Obamacare exchanges in King v. Burwell. The Supreme Court did not issue an opinion today, but within the next two weeks Congress may be presented with a dilemma: What should it do about the approximately 6.4 million people who would lose subsidies in 34 states?

The recently announced proposed rate increases for many insurers offering health insurance through the Affordable Care Act (ACA) exchanges have brought renewed attention to instability in health insurance markets and the long-term sustainability of the exchanges – apart from disruptions that would ensue if the Supreme Court rules in King v. Burwell that the ACA does not allow premium subsidies in states with federal exchanges.

Republican leaders are coalescing around a plan to extend the health law’s tax credits for as long as two years, while repealing other parts of the law, if the Supreme Court invalidates the credits.

The high court is expected to rule by the end of June on whether to restrict the 2010 law’s tax credits—used by low- and moderate-income consumers to help pay their insurance premiums—to the handful of states that opted to set up their own insurance exchanges.

The Obama administration cannot account for nearly $3 billion in subsidies paid to insurance companies in 2014, according to a government watchdog.

Those dollars are untraceable because the Department of Health and Human Services (HHS) did not have “effective” methods to do so, according to a report from the department’s inspector general.

Congressional Republicans will move to temporarily continue health care subsidies for millions of people if the Supreme Court overturns the aid, according to plans discussed Wednesday in the House and Senate.

In addition, the GOP proposals would dissolve many of the basic requirements of President Barack Obama’s health care law, including mandates that most people buy coverage and most companies provide it to their workers, Republicans said. Eventually, they hope, the entire law would be repealed.

In a recent interview with the Des Moines Register, Hillary Clinton outlined several elements of Obamacare that she said she would seek to change as president. Her proposals illustrate how the fiscal impact of the law could increase significantly from what was expected when the legislation passed in March 2010.

Among the things Mrs. Clinton cited was “how to fix the family glitch.” In short, if an individual qualifies for “affordable” health insurance through an employer, that person’s family will not qualify for federal insurance subsidies–even if the employer does not offer family coverage or if family coverage is unaffordable for the household.

Millions of people are waiting anxiously for the Supreme Court to decide the fate of President Obama’s health care law with a ruling this month on health insurance subsidies. But David M. King, a plaintiff in the case, is not among them.

Mr. King, 64, said recently that he was reasonably confident he would prevail in his challenge to the subsidies, a central element of the Affordable Care Act.

“We have a good chance of winning,” he said in an interview at his home here.

Despite White House veto threats, the House is ready to vote to repeal taxes on medical devices and kill a Medicare advisory board that foes say would ration health care as the chamber aims its latest whack at President Barack Obama’s health care law.

Thursday’s votes were slated a day after top House and Senate Republicans briefed rank-and-file GOP lawmakers about their plans should the Supreme Court annul federal health care subsidies for millions. Under the tentative House GOP proposal, states could design their own plans for funneling federal health dollars to residents and drop the health law’s consumer protections, such as guaranteeing that family policies cover children until age 26.