It appears that this router -- and I think a good many others, as well -- have an idle timeout that times out internet-lan connections. I traded some emails with the Xsense guys, and they confirmed the timeout on their router -- and said that there was no way to either disable it or to change the timeout period.
Fortunately, the ssh guys have this figured out already. By opening /etc/sshd_config in your favorite editor (sudo required), and adding the line ClientAliveInterval 120, you set the ssh daemon to "re-awaken" the connection every 120 seconds, which is enough in my case to keep the connection active and prevents the ssh session from hanging. Read the man ssh page for more. I don't know if Apple's AirPort Base Stations also have an idle timeout like this, but there's no reason to think that this hint wouldn't work with them if they do.

