They might have said you would walk out the door, but the reality is that they would have been firing you and that would have been a whole can of worms for them.
Except not really. It would have been "So, sue us." It's not even clear that you would have a case; you're just having to sign an agreement that already applies to all current employees. In any case, is it really worth spending 10s of K$ to fight them?
I interpreted CyberDildonics' comment differently. It may be structured as "sign or walk out the door", but that sounds like "sign this or quit." But in truth, it's "sign this or we fire you".
Depending on local laws, this may make you more (or less - I am not a lawyer and have no experience in this field!) likely to get unemployment benefits.
You may well be eligible for unemployment under the circumstances but that's probably a fairly minimal percentage of what you were making as a mid-level engineer or marketing person.
The point is, the company wants you to quit and quitting is different than being fired. Unemployment benefits, severance, insurance, visas etc are all things that may be different if the company fires you. When they say 'there is the door' they are misdirecting people from the fact that the real alternative choice will cost the company a lot more money.
