You don't need permission or something from Cox to use your own equipment. For the modem, you just buy one on their list and contact them with model/serial number and HFC/CM MAC address to provision it for service. Then you just connect the modem to the router's WAN port and you should be good to go. This can only be done if you don't have wireless TV boxes or phone. If you get the gateway for free, you can put it in bridge mode and use your own router.
Before doing that, take a look at your signal levels from 192.168.0.1 > user;admin password;password > Connection > Cox Network.