The egg became a Christian symbol early on, and in fact dates back to Mary Magdalene, who was seen as an equal of the Apostles, as well as a minister to them. Here is the old story is told (and I do believe this was the end for Pilate, as told here):
Her travels eventually took Mary Magdalene to Rome, where because of her family's standing she was able to obtain an audience with the Roman Emperor, Tiberius Caesar. Her purpose was to protest to him that his governor in Judea, Pontius Pilate, and the two high priests, Annas and Caiaphas, had conspired and executed an innocent man, namely our Lord Jesus Christ.
According to the tradition, everyone visiting the Emperor was supposed to bring him a gift. Rich and influential people, of course, brought expensive gifts whereas the poor offered whatever they could afford. Mary Magdalene took an egg to the Emperor's palace and handed it to Tiberius Caesar with the greeting: "Christ is risen!"
Tiberius Caesar, naturally, could not believe what he heard and responded to her: "How could anyone ever rise from the dead? It is as impossible as that white egg to turn red." While Tiberius was speaking these words, the egg in the hand of Mary Magdalene began changing color until it finally became bright red.
Thus the Pascha greeting -- in universal Christendom, both East and West -- has ever since remained "Christ is risen!" and it became traditional for Christians throughout the world to color eggs in red.
Mary Magdalene then went on to explain to Tiberius Caesar that the now-red egg symbolized life rising from a sealed chamber, a symbol that would have been understandable to a pagan Roman.
Caesar heard the formal complaint presented by Mary Magdalene, and also had received reports of soldiers under Pilate molesting and killing civilians in Judea. For this Pilate was exiled to Vienne in Gaul where he died an unpleasant death. Interestingly, Pilate's wife Procula Claudia who had a dream about Jesus the night before He was brought before her husband for trial, had become a very pious and devout Christian, and died a saint of the Christian Church.
The tradition of the red eggs at Easter has been a tradition handed down among the Orthodox and the Eastern Catholics to this day. Here are some modern day images of this old tradition:
2000 years later, Eastern Churches remembering The Magdalene's red egg gift...