No, there's no chance. Anyone who says there's a chance doesn't understand American politics at all. The government has been shut down 18 times in the past (primarily by Democrats); the government has never defaulted. The shutdown and the debt ceiling are disconnected issues.
The government won't default. Period. Boehner has already publicly said that he would bring the debt ceiling to a vote in the house even if a majority of Republicans weren't going to vote for it, and in that event it will certainly pass. The attempts to get additional spending cuts before they actually have that vote don't need to succeed for the vote to come. The chance is 100%.