Place the red cabbage and beets in separate small pots and cover with water. Bring to a boil, reduce heat to low and simmer for 20 minutes. Strain the liquid and cool.
Add 1 tablespoon of lemon juice to the cabbage water to get violet eggs or 1 tablespoon of baking soda to get bright blueish green eggs.
Allow the beets to soak in their cooking liquid (off the heat) for 1 hour to get red, or use immediately for a rust-ish purple color.
The remaining ingredients do not need to be boiled or simmered. You can just place the combinations in separate bowls.
Place the egg whites in the color mixtures to dye. The egg whites will begin to absorb the color immediately and will continue to darken and become more vibrant over the next 20 minutes.
To dye whole, unpeeled eggs, add 1 tablespoon vinegar to each color. The eggs will absorb the color after about 10 minutes, and will continue to darken and become more vibrant up to 24 hours in the liquid. If you will be soaking the eggs overnight or several hours, cover and refrigerate them.
Note: to get raspberry or blueberry juice, defrost 1 1/2 cups of frozen berries and mash through a sieve or puree in a blender and strain.