They are considered essential. That means they have to work, but not be paid.
https://time.com/7329683/government-shutdown-flight-delays-c...
I think thats due to the 27th Amendment [1]
> No law, varying the compensation for the services of the Senators and Representatives, shall take effect, until an election of Representatives shall have intervened.
can't change (or stop) congressional pay until an election. guess it's a double-edged sword they can't give themselves in immediate pay raise, which I think was the point of ratification in 1992, but also can't cut their pay for failing to pass a budget.
[1] https://constitution.congress.gov/constitution/amendment-27/
... But they're hurting for recruits in a big way so even at their size their negotiating position isn't as strong as they might want.
What's more, even if you could, who is going to take a job for which they will not be paid as long as the shutdown continues?
(Current Airline Pilot here, definitely NOT in favor of privatizing ATC, but historically breaking things on purpose is the usual path politicians take to privatize)