I also had to search first to get an idea how to construct an egg. I have oriented me to the following geometric solution.
And that can be redrawn in the same way in a sketch in the FreeCAD Part Design workbench.
This egg shell can also be used for 3D printing. Therefore I have meshed the shell with 0.4 mm thick shell elements (Netgen) and used the integrated PLA material properties in the FreeCAD FEM workbench. And then you can just use the analysis type “Frequency” to calculate these colorful Normal modes.