May I ask you where did you get the info that binance is supporting ETZ?
I just went to their official website and find some info about the fork. I just learned that binance has not yet announced their support to ETZ. Yes, I transferred my eth to binance and I'm not sure if I can get some ETZ. My fault for not having some research first. I should have left it in my coinomi, if I know.
I found out on Binance's twitter account but did not find the info, seems like hoax. I just found the info on the Etherzero website that they listed Binance
https://etherzero.org/index.html?id=2#Exchanges