I've always been in an "at will" state (Arkansas) and my more recent firing was without cause and without notice. But, since it wasn't for cause, I was/am eligible for unemployment insurance.

I'm not claiming, because I can't actually accept any job offer, due to the constraints of familial duties, and Arkansas requires you accept an offer when you on UI. (I think there are conditions where you can reject, but I've not read the statues, just the guidelines published by the relevant Arkansas government department, which aren't that explicit.)

This is false. You cannot be fired without reason, you can be fired for any legal reason which will vary from state to state, which may be more strict (for employers) than federal law.