Yes it will be a problem for the eggs if they get x-rayed. I used to work at Heathrow on the x-ray machines,amongst other things,and we were instructed not to deliberately x-ray any animal. Most of the time it was cats and dogs,so they came out of the cages,the cages were x-rayed,then the animals replaced after.
My only suggestion would be to take a very small torch,so they can x-ray the container and candle the eggs.... Ree