Because eggs put up a tough fight to keep the color out.
First, some hard-boiled facts. Baby chicks come from fertilized eggs. (But don't worry -- eggs you buy in the grocery store aren't fertilized.) Everything about an egg is designed to protect the growing chick.
The next time you peel a hard-boiled egg, look closely at the shell. That filmy skin between the shell and the egg -- the one that sticks to the shell as you peel it off -- is the baby chick's first layer of protection.
There's another filmy membrane next to the egg's white part, and there's a thicker membrane around the yolk. All these barriers work as an umbrella to protect the chick from bacteria and other organisms. (An eggshell has more than 6,000 pores that microscopic bugs can get through!)
Dyes -- like food coloring or natural colorings from plants -- all work basically the same way. Their colored molecules hook up with molecules of the material to change its color.
If the inside is colored, it could be because of a crack in the shell. Or if your dyed water is colder than the egg, the color has a better chance to sneak inside. But if it all goes right, dyes normally don't color the hard-boiled whites -- the shell membrane works hard to prevent the color from soaking through!