X-rays or other scanning devices that would be used through customs will not magnetize your watch. Even metal detectors, which use magnetic fields to detect the presence of field disturbances, us AC current so won't magnetize your watch.

If this were the case there would be LOTS of destroyed hard drives imported from Japan and China.

This seems to be coming up a lot. I might post up a thread explaining a bit about electromagnetism and how it relates (or doesn't) to magnetism. Any interest?

Yes, that would be an interesting read. Please post a lengthy explanation.

My finely tuned ears might be catching a bit of sarcasm....
Well I promise I won't make it too lengthy. I'll get to work on a little something in the next couple of days. A little physics vis a vis watches.

so an actual X ray machine at the doctor's office should be fine too?

This seems to be coming up a lot. I might post up a thread explaining a bit about electromagnetism and how it relates (or doesn't) to magnetism. Any interest?

If it's just an x-ray then yes, but the doctor may want you to take it off if your arm / wrist is being x-rayed.