Most military jets can be hot refueled on the ground, with engine(s) running. Some fighter jets (F/A-18s etc) can hot swap crew on the ground, with one engine running.

With the F-35 and other jets, its possible to swap crew and perform minor maintenance with the jet electrically powered up and cooled with auxiliary air supply, although the engine is shutdown and then restarted. This actually saves significant time on the pre-takeoff procedures. I would think the Lockheed Martin engineers could also perform diagnostics after a failure, without interrupting power or restarting the sub-system.

I'm actually curious what possible technical reason would prohibit them from running for 48+ hours (or indefinitely). IE some kind of garbage collection issue. I'm unaware of any reason they should ever have to be rebooted aside from software upgrades and curious why it might be necessary (technical reason). Unfortunately, from the article: "But the Air Force did not go into detail about when the instability in the radar system occurred..."

Edit: It is mentioned elsewhere in this thread that it is a timing issue, see: http://breakingdefense.com/2016/02/bogdan-predicts-f-35s-for...

Have to shut the engines off at some point. Any vehicle "reboots" when you shut down the power.

It is certainly possible, but not normal. Normally the engines are shut off during servicing and the aircraft is hooked up to a ground cart to provide limited systems functionality.