Hmm, a few possibilities come to mind, but they are all rather unlikely. 1, there could be some type of electrical feedback or ingress on the Ethernet ports for the LAN. Maybe your using shielded Ethernet and the connection points at the ports are transferring that current picked up by that shield. Or maybe you have a bad power supply and its running current through the Ethernet from one of the PC's into the router. It could also be some intermittent connection issue that isn't effecting the wireless, since its designed to deal with temporary lapses in connection? That wouldn't explain why it would need a reboot after though. The only other thing I could figure is maybe the wireless is a false positive? Maybe your connecting to someone elses wireless so your not experiencing the symptoms? Either way, we need more data to be sure. Try this:
1. When the wired goes down, make sure your online (try multiple sites) and offline on all wired.
2. Turn off the wired connections one by one, checking a wired connection as you shut each other wired connection down.
3. If all are still down, disco all wired from router ports. Power cycle just the router. (With the router off, make sure all your wireless is offline, this will confirm your wireless connection is to your router)
4. Plug in one ethernet at a time and make sure online. Try to wait as long as possible, depending on the time you have to work on.
5. If it still happens. Try to check the router logs, see if it gives any error which might tell us the cause.