Word to the wise! NEVER! EVER! EVER! use booleans! Stay away from them!

Start with the egg. Erase various faces by switching to Face mode. Delete a good part of them by selecting them and hitting delete. Switch to vertex mode and shape the vertices into the shape that you want by selecting them and using the move tool to move them around. Easy!

If you want to get a thickness of the egg form inside the hole. Switch to Edge mode. Select the edges that make the hole. Got to: Edit polygons>Extrude edge. This will extrude edges from the edges that you selected. Switch to the move tool and move the new edges inward inside the hole.