I've also had trouble logging into Twitch a few times over the last year on Firefox, but the same is true for Paypal. Both of them don't work in a private window without any addons either, and at least for Paypal changing the user agent didn't help. Twitch works fine If I'm already logged into Twitch, same with Paypal. Just the login fails for some reason.

There's other payment options, and I seldomly watch streams anyway.