Place water, vinegar, and the selected vegetable, roughly cut if necessary, into a medium saucepan.
Bring to boil over medium-high heat. Reduce heat and simmer for 20 minutes with the lid on.
Remove from heat. Allow to cool for 10 minutes. Pour the dye through a fine mesh strainer.
To dye eggs, place two boiled eggs in a jar. Pour the dye or coffee over to cover. For the deepest coloring, place the lids on the jars and refrigerate overnight.
Carefully remove the eggs from the dye and transfer to a wire cooling rack. Allow to dry completely before touching to avoid disturbing the dye.
Notes
Eggs dyed naturally will differ in color and intensity from those made with chemical dyes. Half the fun is seeing what they look like after they emerge from the dye!