In the news, the reporters have been talking about "essential" (real quotes, not scare quotes) government employees being required to work without pay by these two statements (paraphrased).
All employees will have to forgo pay during the shutdown.
Essential employees will be working during the shutdown while non-essential employees will be sent home.
One example of such claims: https://www.kcra.com/article/what-a-partial-government-shutdown-could-mean-for-you/25648684
I remember the Thirteenth Amendment banning "slavery and involuntary servitude" and this is in fact so powerful that the federal government cannot force private practitioners to take Medicare at all.
I again remember the ability to draft for reasons of national security, but I know of no case where that drafting was not with full pay according to the normal government pay scales.
But even if the government is really out of money (which it isn't; during a shutdown it should be running a tax surplus*), I can't imagine how this leads to requiring people to work without pay. My normal assumption would be you print unbacked money if you have to (and deal with the inflation later) to pay the troops.
But they say the employees will be required to work without pay. How?
*They say the shutdowns cost the government money. The net loss is due to the almost universal practice of paying all the government workers after the fact.