The internet is full of amazing things, including blogs on how to make natural egg dyes. We tried it, and - it works! Here's how we did it:

Put your ingredients (like turmeric, spinach or beets) into a pot with 2 cups of water and bring to a boil. Cover, reduce heat and simmer for about 10 minutes (the spinach takes longer). Strain the liquid and discard any solids. Stir 1 teaspoon of white vinegar into the liquid and let cool to room temperature before dying your eggs.