I would say no, for two reasons. The first is that I am against the idea special privileges and immunities arrogated to the state. I do not believe it is at all proper that Federal employees should receive special treatment over non-Federal employees and be given immunities not enjoyed by other citizens.
The Privileges and Immunities clause has to do with one state legally discriminating against out-of-state US citizens. It's pretty much the most basic implementation of the supremacy clause, though it is indeed its own constitutional provision. (It was also made just about irrelevant well over 100 years ago). But it has nothing about saying you can't evict federal workers furloughed in a shut down. Granted, there is much debate about it given the paucity of primary source info and precedent that would have recounted said info in prior decisions.
What
might have something to do with a shutdown is whether a blue state could for example say that people who have lived in that state for X years cannot be evicted if federal workers in a shutdown, but people who moved there recently and therefore have lived in the state for less that X years could be evicted despite being federal workers in a shut down. Just an example. The clause is not about a federally imposed restriction on states about what they can do with federal workers as opposed to non-fed workers. It's basically about state-state petty legal warfare.
Or, for example, a state law saying citizens of X state cannot travel through their state. Or cannot live in it at all.
Now maybe you want to try to base it in a lack of authority to pass a federal law telling states they cannot allow eviction of federal workers in principles of federalism or perhaps Intertate Commerce (which is now basically as wide as one wants it to be), but it's not privileges and immunities.
Second, what about the people to whom the Federal employees owe money? I think if that is done, I do not think many landlords are going to be falling over themselves to rent to Federal Employees. Let us pretend, for the sake of argument, that I am a landlord and I own a rental home that I still owe a mortgage on. Let us further pretend that I am renting to a Federal Employee who cannot pay me my rent, and under normal circumstances I would need to evict the person and get in a new renter ASAP. Why should my credit be ruined because I cannot afford to pay that mortgage because I am not receiving the rent, and am still responsible for maintaining this property of a tenant at sufferance?
Well, if we ignore Bob's idea and instead pay them, that won't be an issue.
Or, Bob's idea could be supplemented not by saying pay the federal workers during a shutdown but rather with a provision that says you can't demand the portion of a mortgage payment assignable to a renter of a person who rents to a federal employee, and now you piss off the banks. (Though, they probably could absorb that a lot better than federal workers and their landlords could absorb the alternative).
Again, the problems you might run into with stuff like the latter would have to do with a different clause than P&I.
Really, they should be paid during a shutdown. They didn't do a damn thing wrong. Usually, they're caught up with a stupid take-hostages-issue-demands situation. They're now caught up in the dumbest version yet.
All they'd have to do is tweak law/reg about what counts as an "essential service" such that this becomes a thing.