In the event that there is such a stalemate between the elected representatives within the house and within the senate that the government shuts down, it is likely that the true wishes of the electorate are not being satisfactorily expressed. Therefore, it will likely prove beneficial for the voters to search for different representatives.
In the event that a budget is not agreed upon and the government will be forced to shut down, an election should be held for ALL seats in the House and in the Senate. First, a vote should be held to recall each representative, so that the voters that they represent have an opportunity to express a lack of confidence in their abilities, and then a vote for a new representative if the incumbent is recalled.
Only one such election may occur each year.