How to get Messages to connect to Facebook chat?
Your work machine likely blocks port TCP 5222. Or both work network and iPhone mobile operator block port TCP 5222 or chat.facebook.com. Please ensure those are opened on any in between firewall.

Even when the network provider does not actively block port 5222 there can still be issues in the network as this discussion of 81 posts in length shows.

To be absolutely sure that the issue is network blocking, you could try to setup a VPN connection on the faulty work computer. For instance to your home computer or use the free VPN demo from usaip.eu.