For screen shares or voice calls, Screenhero tries many different things to establish a peer-to-peer connection, so it could use nearly any port. The exception is if connection needs to be routed through our relay server (which it will do only if it's unable to mediate a direct connection) -- in that case it will use port 443/UDP.
For direct connections, it will use UDP on whatever port the OS automatically assigns it. Allowing outbound UDP sessions (including the return traffic for those packets) should be sufficient. There are other small things that can throw it off, such as if your firewall rewrites the port numbers of outbound traffic, but those are fairly rare.
If you have any questions, please contact us!