First, it's bullshit because it's unconstitutional. According to the 27th Amendment, Congress cannot cut (or raise) their salary within a term. Any change in pay will not take effect until the next term. Let's take a look:
No law, varying the compensation for the services of the Senators and Representatives, shall take effect, until an election of Representatives shall have intervened.
I know many "Constitutional Conservatives" and "Originalists" like to say that the Constitution is not a living document. Article V, the one about amendments, the Founders were only kidding. Many act like only certain amendments after the first ten count. Here's where knowing a little history comes in useful. This amendment was part of the Bill of Rights. Twelve amendments were passed around in 1789. Only ten were ratified by enough states to make it into the Constitution at that time. This one was ratified by seven states.* This is not a new idea. This comes from the Founding Fathers themselves.
There might be some wiggle room to suspend their pay, but they'd get the full amount after the deal was made.
The second reason that it's bullshit is that, well, it's bullshit. Even if we cut or suspended their pay, they make over $170K per year. How many of them would really hurt if they had to go without pay for a couple weeks. Very few. Meanwhile they would dig in their heels (or keep them dug in) shouting for all to hear that the gridlock is entirely the other party's fault. Democrats will continue to claim that bipartisanship and negotiation means both sides give a little to find a middle ground. Republicans will continue to claim that bipartisanship and negotiation means Democrats agree to vote for the Republican program and agree to take most of the blame for the unpopular parts. In the end we would get almost the exact same deal that we're going to get anyway. The Democrats will give far more than the Republicans and both parties will loudly proclaim "never again." The only difference will be that the deal might come a few days earlier.
* The last of the original twelve was a bizarre formula for increasing the size of the House as the population grew. If ratified, The House would have about five thousand members today.