I was laid off and was eligible for ei, you will be too but I would recommend filing immediately. Even though you don't have roe you can hand it in later. It takes a few weeks for ei to get approved and you will have the 4 week severance added on to your 2 week waiting period so plan on about 6-8 weeks of non payment.
Ei I believe is only non-applicable if you quit for no legit reason or fired for just cause, since they're paying you severance obviously there's no legit reason so I wouldn't see why you're not eligible. Google ei application and start tomorrow morning, or even tonight if you read this in time. Good luck and I know how it feels. Sorry to hear
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