I think the trick to the claw game is just to aim very carefully. Position the very center of the claw (in between the pincers on part one of the turn) over the egg you want, and then on part 2 of the turn, press stop when the claw is exactly above the egg.
Sometimes I think I've positioned it right, but it still turns out to be a little off, and the egg drops on the second screen even after the claw was shown to pick it up. If you don't see a musical note when the claw drops to pick it up in the first screen, it will drop the egg.
Just keep trying, I guess. ^_^