The eggs have to make a symmetrical pattern in the box otherwise something doesn't feel right.

They used to do boxes of 15, which worked perfectly. Now it's either 6 or 12, both of which potentially require you to adjust the number of eggs you eat to get a symmetrical pattern.

It is both necessary and sufficient that the number of holes in the egg box should be an odd number.

Nine and fifteen work really well. All the other odd numbers are either too big, or negative, or prime, which would be impractical.