I'm so sorry this is happening to you! I would talk to an employment attorney in your area because it seems as if they are trying to build a case against you to eventually fire you. You should protect yourself going forward. Most employment attorneys will do a free consultation and you can find one here on Avvo. Good luck!
If you are "at will", you can be terminated in New York for any reason absent discrimination, etc. For Unemployment Insurance purposes, you will not be entitled for benefits if the DOL determines if your quitting was without compelling cause.