A government shutdown on Tuesday is still a very strong possibility. The Republicans controlling the House in Congress and the Democrats controlling the Senate are still at odds over a short term spending bill that would allow the government to continue its operations. The House's version would defund the Affordable Care Act, while the Senate is prepared to pass a "clean" bill that removes the defunding plan. House Speaker John Boehner said the House will not accept the Senate version, but there are still a couple of days for Congress to figure out what to do.
The government shutdown, if it happens, won't even affect Obamacare. There are certain essential portions of government that continue on, like the military, Social Security and Medicare, because their funding is manadatory and not tied to annual resolutions. Obamacare is one of those.
All members of Congress should do what is best for the country, not what is best for their party, and it is hard to imagine any scenario that justifies a government shutdown.