1. neither support 802.1X as far as I'm aware
2. you can't "block" them - you'd just put them in a "deny" role, so they would be connected to the SSID but can't rx/tx any data and hopefully the owner of the device will not longer try to access the SSID
So in case you use a PSK SSID just enable MAC based authentication and use the NAC for authentication.
Add a rule with "device type group" gaming and a deny profile.
Add another rule which must be after the deny which allows all other devices.
If the gaming device connects to the SSID the NAC should authenticate it with the deny role and other devices get the allow role.
I've tried it and it works for the PS4 but unfortunately the XBOX One is identified as device type "Windows".
So you'd need to open a ticket so the GTAC could try to add a better fingerprint.
They have done it for me with the right data within a day for some other devices.
Here a link what data they need from you....
https://gtacknowledge.extremenetworks.com/articles/How_To/NAC-Troubleshooting-Tips-Debug-Methodology...
-Ron