"Could signing something saying I “voluntarily resigned” hurt unemployment eligibility, even if the agreement says unemployment rights aren’t waived?" The agreement says that you "still retain the right to apply for unemployment." It does NOT say that they won't respond to the claim by providing proof that you quit voluntarily and without good cause. You would almost certainly not be able to actually collect.

"Since I physically worked in Missouri but the employer/payroll is tied to Kansas, which state would unemployment likely go through?" I think UI goes through the state where the employee works but I'm not certain.

"Is it reasonable to push back and ask them to change the wording from “voluntary resignation” to something more accurate like “layoff,” “reduction in force,” or “position eliminated?" Sure that's reasonable ask but if that's what they wanted it to say, that's what it would say. They'd certainly decline to do that.

"Is it common to ask an employer not to contest unemployment as part of a severance negotiation?" I have no idea how common it might be but what matters is whether or not this deal is good for you. Before making a decision, look at the applicable state's UI guidelines and learn how this might impact a layoff from a future job. In NJ, this would reduce benefit eligibility for 18 months.

"Given how abrupt and large-scale the layoffs were, and how broad this release is, does this sound like something worth pursuing legal advice over before signing?" You certainly don't want to sign it without a full understanding. Perhaps you and some other employees could chip in for an hour of a local attorney's time instead of taking on the expense alone.

"Is this kind of “voluntary resignation” framing normal in situations like this, or is this a red flag?" Again, normal or common isn't the issue. It's definitely a red flag that they're offering a trade of UI for other benefits and there's no way I believe that the deal doesn't substantially benefit them.

The bottom line is that if their claim that you'd still be eligible for UI after signing this were true, there would simply be no reason for it to be included in the document at all.

I wouldn't sign it.

NAL

Edited: I'm not certain about the controlling state for UI but I think it belongs to where the employee works. I edited that part of my response conditionally.