The "server" that Gizmo is referring to when it says it cannot reach the server is typically your computer, not JRiver's servers. The PC that Gizmo is controlling/playing music from is a library server serving content to Gizmo. When gizmo says "cannot connect to server," what it means the vast majority of the time is that it cannot establish a connection with your computer within your LAN.
That could be happening for lots of reasons (lack of fixed IP addresses, firewall issues, wifi connectivity, you name it).
The only time Gizmo pings JRiver's servers is when it starts up (or changes servers) to resolve the access code, and you can sidestep that entirely by entering your computer's ip address (+ ":52199") into Gizmo instead of an access code. It's less convenient for you if you use Gizmo outside of your home network, but if your uses are LAN only you should get the same functionality (and likely the same connectivity issues you're currently experiencing).