What paints can be used on eggs?
Food coloring would be the easiest, most colorful and most common way of coloring eggs. 

Alternately and perhaps more traditionally there are a number of natural dyes that come from food stuffs: purple cabbage, red onions, yellow onions, beets, teas. Typically these are chopped and boiled in water with a little clear vinegar in it until the color goes into solution. Then it works like any other coloring... dip, brush, or soak the eggs in it to give them color.