To be in a delicate situation; to be on the edge of danger or ruin.
An 'eggshell' is the thin, white outer coating of an egg. The shell is very thin and breaks easily. So if you are 'walking on eggshells' you are in a situation where you could break something (or get in trouble or ruin everything) very easily.
When a bald or balding man, seemingly unaware of hair loss occurring on the top of his head, continues to grow out the hair on the sides and back of the head, thus ornamenting his barren follicles with a flowing skirt of, often disheveled, locks.