Shiny Server is implemented in node.js but not designed to be extended directly with your own node.js code. However, what you could do is put another server in front of it, written in node.js, go, or whatever, and proxy requests intended for shiny server to your shiny-server instance. That's a really nice thing about HTTP servers/proxies, they compose very nicely.
Just be aware that your proxying logic ideally will support not only HTTP but also WebSockets. Or you could disable websockets in shiny-server, then you'd only need to proxy HTTP--this would just be a little less efficient but otherwise should work OK (it's what we currently do on ShinyApps.io).