Code attached "due to popular request" (well, one. 
)
The "manual" version just moves one motor at a time. It was more for tweaking the mechanics. Search in Arduino here for the nnnC library - I wrote that.
The "1" version draws what the video shows and is hardcoded. The intention is to include a library of patterns which are selected with some simple input mechanism (so the USB/Laptop is not needed)
Neither program is complete or written to be educational (too few comments, structure/variables).
EggBotTest.ino (1.43 KB)
EggBot1.ino (3.66 KB)