First take a look at this: AndEngine Tutorials
Be sure you have the latest AndEngine, I'll advise you to use AnchorCenter branch, where by default all your sprites will have thier anchor center in the middle of "their body". After that just create 2 sprites, the Chicken (C) and the Egg (E) both with the same position but with different zIndex where the C will have a higher one, so that the E will be behind hidden. Then override the C sprite onClick method (the C could be a ButtonSprite instead of a simple Sprite). To move the E you could use an EntityYModifier with the desired offset.
This is the main idea ;)