Rotten eggs stink, so no one wants to be a rotten egg. Children often say this to get their friends to join in something, especially to jump into a swimming pool (pond, lake, ocean, etc.) When one says it, all of the others run as quickly as possible to get into the swimming pool and not be the last one in.

in lithuania we use this expression too

it is a playful expression to start a race to a specific spot. used mostly by children. in english we would say in a group "last one in the pool is a rotten egg" or something playful like that

An individual who turns out to be rotten, as in You can't trust him—he's simply a bad egg. An early definition appeared in The Atheneum of 1864: “A bad egg ... a fellow who had not proved to be as good as his promise.” In contrast, the schoolyard saying Last one in is a rotten egg does not have any special significance other than as a way of urging others to join an activity, jump in the water, or the like. http://idioms.yourdictionary.com/bad-egg