This is false. If a company makes you sign a statement that you are quitting instead of being fired as a condition of getting severance, they are committing unemployment insurance fraud and are subject to civil and criminal sanction in most states.

(The point of trying to make employees do this is to avoid claims against the company's unemployment insurance account in the state. How UI works differs from state to state, but in all instances claims against UI increases the company's ongoing UI expense.)

Yep, in the case of the lump sum you're not being paid after you're given the lump sum. I was laid off in 2014. Got 4 months in a lump sum when I went out the door. There were no problems applying for unemployment.

I went with the latter because it paid better.

For a while I was always stressed out about background checks, because it showed that I was working two jobs. (I got a job to replace it almost immediately, but continued to get paid by the old place.)