You will still need to rely on what the platform has available for you. So on Android you would still need to have a BroadcastReceiver or at least a Service which has the SignalR connection, otherwise it will not survive if the application is closed.
iOS and Windows is a bit of a different story, there you are not allowed to have stuff running in the background. So here you would probably need to use push notifications to notify of new messages. Although on Windows Phone you can have Scheduled Tasks or also called Background Agents here I think you can do something every 60 seconds at most.
On iOS you are out of luck you simply can't do stuff in the background unless you are using the built in Location Services or something like that.
MvvmCross does not do anything magically, nor does it allow to create a service which does something the platform does not provide.