Can it be unhealthy to go without eating for two full days, then eat little for the rest of the week?

Yes. Nutrition is very important to health. The body needs calories to do it's functions. The brain needs energy to think and perform. Without good nutrition, we weaken our immunity and are subject to infections and other maladies. Skipping meals is not a good way to stay healthy. Neither is over eating. Eat a proper diet, balanced with the food groups and exercise. 3 meals a day with snacks!