From your description it's not entirely clear what your setup is. For example, are you using RStudio Server open source (or the Pro version)?
My completely wild guess is that perhaps you are blocking websockets on your network. By default Shiny uses websockets to communicate with the browser, but this can be changed (see for example this). (You could possibly inspect the JavaScript console for errors to confirm this.)
By the way the localhost address where shiny is listening is normal behaviour, and you should not need to modify this at all.