Current beliefs of Richard + the EeePC forum are that the problem is either:
a) The X-ray machine and/or a metal detector at the airport wiped the flash, so it won't boot because it contains no operating system or programs. (EeePCs are ridiculously cheap at least in part because they contain absolutely no drives - they don't even have a hard drive.) This is unlikely, because X-rays aren't supposed to affect flash memory, and if it did, our cameras and my phone/PDA would have been upset too.
b) When my bag fell on the floor during the train journey to Gatwick, the shock of the fall dislodged some internal connection to the flash or the RAM, so it won't boot because it can't see that. This seems rather more likely, and if I was the sort of geek to carry a micro cross-head screwdriver everywhere I went, it'd be easy to open up the case to find out. Except I'm not that kind of geek, and especially not when flying and having to conform with annoying "security" regulations.
With this being New York and The City That Never Sleeps, my lack of screwdriver could be easily remedied by a trip to the 24 hour hardware store that exists not terribly far from here according to the Yellow Pages. Except my body thinks it's 4 in the morning, and I only had 3 hours sleep last night and 4 hours the night before. Much as going to a 24 hour hardware store in the middle of the night sounds like A Thing That Really Should Be Experienced at least once in a lifetime, I'm not sure that now is the time to do so. I think that a hot shower and bed would be a much better idea.