Animals That Lay Eggs Don't Have Belly Buttons

Verdict

A belly button is specifically for use with an umbilical cord. Egg-laying creatures have no need for umbilical cords. I rate this as true.

Most animals lay eggs.
A select few, the Placentals, utilize a placenta -- an interior fluid-filled sack -- and, by proxy, an umbilical cord prior to birth.
A belly button is the hole to which an umbilical cord is connected during gestation -- the time when a fetus/baby is growing -- in placental mammals. The cord connects mother and child for the purpose of feeding, blood sharing, etc.
Thus, if you lay eggs, you don't need to be attached to them via umbilical cord. If you didn't have an umbilical cord, you won't have a belly button.
There are three kinds of mammals -- one that uses umbilical cords and two that do not.
Marsupials have a pouch in which they keep their young (opossum, mole, quoll, wombat, kangaroo, etc)
Monotremes lay eggs like most creatures (platypus and echidna)