There are lots of edible, natural dyestocks that can be used to color eggs. Listed below are those with which we've had the most luck over the years.
The results will vary from batch to batch and from year to year, depending on amount of dyestock used, length of cooking time, and various other imponderables, possibly including (who knows?) the phase of the Moon. The colors you get may not be predictable, but they'll certainly be beautiful.
Most natural dyes are heat-applied; realistically, this means that you add the dyestocks while you hard-boil the eggs. Dyeing eggs is, of course, a controlled chemical reaction in which loose radicals in the dyestocks bond with the calcium molecules of the eggshells, and heat facilitates this bonding. As always in the pagan world, it all comes down to relationship.