I live in NY State. I was just given notice that my position will be terminated. The termination agreement states "This termination will be considered an irrevocable resignation by Employee, which has been accepted by Employer."
If I sign that agreement (which is the only way they will give me severance pay) am I still eligible to receive unemployment benefits?
Your severance agreement most probably states that you should see a lawyer before signing. Even if it doesn't, you definitely should. As far as unemployment, it does not matter what the agreement states. Unemployment will make their own determination based on the actual facts and circumstances that caused you to apply. You will be eligible for unemployment if you are being laid off without cause.
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Generally speaking, such severance agreements have been held not to be a bar from receiving State unemployment benefits after exhaustion of any payments that can be considered salary payments. My answer, however, is limited to what's stated in your question. An attorney would need to look at the entire agreement in order to give you a more informed opinion.