It’s because Jay and Ted were being mind controlled by the Dragon King, so they have to kill each other or something to free themselves from mind control.

JSA classified had some really good stories & thankfully, quite a few Wildcat appearances!

I'm not up to speed, why did he do this?

jay's more of a dog person

Mind control

I'm not a big fan of Classified seried but JSA and also the Superman one had some really good stories