The unemployment woman said "Why would she claim you quit when you didn't?" I HAVE NO IDEA.
Well- so she won;t have to pay unemployment.
She's used to bulldozing people and this is just standard operating procedure for her. Here's the thing- you pay into unemployment insurance - it's there specifically for this situation - don't let the fact that she is apparently going to make you jump through hoops prevent you from getting what is rightfully and justly yours.
She'll say stuff on the call to get a rise out of you. Stay calm and stick to the facts (i.e. - "That is not a correct representation of what happened").