The Telegraph is reporting that some Kindle users (one user really, according to the report) are complaining that baggage scanners in airports are damaging the E Ink display in their e-reader. They suggest that the X-Ray radiation may permanently damage the display.
It's highly unlikely that this is a common issue - there are millions of E Ink e-readers on the market (in 2011 alone E Ink will have shipped over 25 million e-reader displays) and many people are taking their e-readers on a plane. If it were a real problem, we would have noticed before. E Ink themselves told us that they're not aware of a single display failure as a result of an X-Ray machine. It's also possible that the failure in the e-reader is not related to the display.
Daping Chu the Chairman of the University of Cambridge center for Advanced Photonics says that it's highly unlikely because those scanners do not have a strong radiation - not strong enough to damage the screen. He does say that it may be possible that static (caused by the rubber belt) may build up and hurt the display.