You are doing it right setting the arm to 90° when the pen is down. I use the “paper under the pen” trick as well.
Make sure your pen tip is in the center of the egg along the ‘long’ axis (from tip to tip).
The pen arm motor height is also ‘critical’ as the pen will move ‘across’ the egg when moving the pen arm from one end to the other. I use a pencil to check this as the line can be erased. Set the pencil tip on the egg with motors off and move the pen arm manually from end to end. If the line drawn on the egg has a ‘S’ shape the motor height is incorrect. (You can also observe the pen arm moving up/down). It takes a few tries to get a fairly straight line.