This is a one of the easier Orthodox and Catholic Christian traditions to explain: The eating, as well as painting and giving of painted eggs, developed as a by-product of traditions about fasting.
New Testament teaching about private fasting, and remembering ‘Christ our Passover’ weekly
It first needs to be said that there is no instruction at all in the New Testament regarding any kind organised fasting, and Jesus’ instruction (below) that if someone fasts then they should not publicise their fasting effectively makes organised fasts impossible without disobeying Jesus’ words.
16 “When you fast, do not look somber as the hypocrites do, for they disfigure their faces to show others they are fasting. Truly I tell you, they have received their reward in full. 17 But when you fast, put oil on your head and wash your face, 18 so that it will not be obvious to others that you are fasting, but only to your Father, who is unseen; and your Father, who sees what is done in secret, will reward you. (Matthew 6:16-18 NIV)
The New Testament also contains no instruction to Christians to celebrate the Passover period when Jesus was crucified and resurrected, and Paul tells the Corinthians to celebrate the Christian Passover weekly with bread and wine in 1 Corinthians 5:7-8
7 Get rid of the old leaven, that you may be a new unleavened batch, as you really are. For Christ, our Passover lamb, has been sacrificed. 8 Therefore let us keep the feast, not with the old bread, leavened with malice and wickedness, but with the unleavened bread of sincerity and of truth. (1 Corinthians 5:7-8 NIV)
European traditions
Nevertheless fasting at “Lent” (the traditional period before “Easter”), developed as a tradition in Eastern (Greek and Middle Eastern) Christianity and passed also to western Roman Christianity. As a result in Eastern Christianity meat, eggs, and dairy were all prohibited during the fasting period of Lent. This led to “pancake Tuesday” or, in French, “mardi gras” or fat Tuesday when all the eggs in the house had to be consumed before the fasting started.
Now obviously chickens don’t stop laying eggs simply because people have stopped eating them. This led to families stock-piling the eggs, and hard-boiling them, waiting for the day at the end of the Lent period – Easter Sunday – when people would be able to eat eggs again. The associated traditions of painting eggs, and egg-hunts for children, are traditions associated with this. All of these European traditions are without base in the Bible, but they are not pagan, and are harmless.
Chocolate eggs were first manufactured in Britain by Frys of Bristol and Cadbury of Birmingham in 1873 and 1875 respectively. Handmade chocolate eggs of various sorts had already been made in various places in Europe by cooks and bakery shops for a hundred years before that, but Frys and Cadbury made giving chocolate eggs as gifts a popular practice – ironically at the same time as observing Lent was becoming less and less common.
The next question probably would be “as Christians, should we have anything to do with this?” – the answer to that is probably that eggs have already ceased to have any real connection with the Christian Easter period anyway. Although the chocolates are still seasonal they are more likely to be connected in the minds of children with school Easter holidays than with anything Christian. What this means is that if parents are Christian and they want to make children aware of what happened at Passover 2,000 years ago, they need to teach that directly – and best done separately from Easter eggs.