Can we implement our own STUN/TURN server to proxy connections to Genesys Cloud?

No. Genesys Cloud does not support the use a private STUN/TURN server nor does it support private proxy servers. Genesys Cloud leverages Google’s global network of STUN (not TURN) servers for faster and better NAT traversal. If you disable them, then you will experience broken or slow NAT traversal during the connection of features like video chat or WebRTC phones.