The tradition of dyeing bird eggs and hiding them at the beginning of Spring predates the Easter holiday. Since bird’s eggs are a sign of the coming bounty of the Spring, as well as an obvious sign of fertility, people were celebrating with eggs and images of bunnies (also a fertility symbol) at the time of the vernal equinox. Many varied cultures had rituals involving eggs at this time of year and decorating them was common.
If you’d like to dye your eggs with natural instead of artificial colorings, consider using foods and spices like people have been doing for thousands of years.
Better Home and Gardens has a wide variety of suggestions for different natural dyes.
The Serious Eats blog also has some great suggestions, as well as describes the “onion skin” technique of boiling the eggs and dyeing them at the same time.