While some congressmen need their salary, most don't. You generally have to be fairly monied to make it that far in politics, meaning that the pay isn't a big deal. Also the lack of pay is something of a hollow threat as in all cases I know of, they authorized pay for employees retroactively after the shutdown.
That aside, if they were furloughed, they'd be prohibited from working meaning prohibited from resolving the situation.
A more effective solution would be to force them to work. Something like in the event of a shutdown they are required to stay in Washington and be in session 12 hours a day, 7 days a week until it is resolved. I think that would be more likely to work.
However, it is all academic since congress would be the ones who'd have to make that law (barring an amendment) so it won't happen.