Yes, cockroaches are edible and healthy to eat provided that they are cooked properly to ensure that bacteria present on the body is killed prior to eating.
You probably consider eating a cockroach disgusting, as in Western society we do not value insects as a food source. But in many cultures, especially Asian cultures, insects including cockroaches are commonly included in the diet.
Here in Thailand, insects are served in many local dishes and are usually fried in vegetable oil prior to eating. Food carts can be spotted with the various types of insects on display, which many Westerners and tourists often gawk at and stopped to take pictures.
But cockroaches are very high in protein in particular, and are also a cheap and abundant source of food in places that may be lacking in inexpensive food choices, especially meats and other proteins.
The locals here in Thailand claim that they taste great, and that they form the basis for a number of comfort foods in the local culture. I have not yet felt adventurous enough to take the plunge and taste a cockroach, but I take their word for it.
Cockroaches do carry a lot of bacteria on the body so it is essential that during preparation they are heated properly to kill off this bacteria. Frying them in oil serves this purpose as well as giving them a crunchy texture to the taste.