I've created a Test class which should help you with your question, you've got several problems in your code.
import java.util.ArrayList;
import java.util.Arrays;
import java.util.HashSet;
import java.util.Iterator;
import java.util.List;
import java.util.Map;
import java.util.Set;
import java.util.stream.Collectors;
    public class Test {
    public static void main(String[] args) {
        new Test().test();
    }
    void test() {
        Set<Food> foods = new HashSet<>();
        foods.add(new Food("a", Meal.BREAKFAST));
        foods.add(new Food("b", Meal.BREAKFAST));
        foods.add(new Food("c", Meal.LUNCH));
        Set<List<Food>> mealPlans = getAllMealPlans(foods, 0);
        mealPlans.forEach(f -> System.out.println(Arrays.toString(f.toArray())));
        System.out.println("--------------------");
        //I think what you might want is a Map from Meal to List of Foods
        Map<Meal, List<Food>> mealPlans2 = getAllMealPlansMap(foods);
        mealPlans2.entrySet().forEach(f ->{
            System.out.println("Meal: "+f.getKey());
            System.out.println(Arrays.toString(f.getValue().toArray()));
            System.out.println("");
        });
    }
    public static Set<List<Food>> getAllMealPlans(Set<Food> foods, int numMeals) { //numMeals unused?
        Set<List<Food>> set = new HashSet<>();
        Iterator<Food> iterator = foods.iterator(); // only instantiate the iterator once(!)
        while (iterator.hasNext()) {
            Food food = iterator.next(); //only call .next once(!)
            List<Food> aList = new ArrayList<Food>();
            if (food.meal == Meal.BREAKFAST) {
                //set.remove(foods.iterator().next()); //why remove it from the set?.. and by foods.iterator() you create a NEW Iterator which is starting at index 0 again .. and check that you are calling .remove with the correct type
                aList.add(food);
                if (food.meal != Meal.LUNCH) {
                    //
                }
            }
            if(!aList.isEmpty()) {//only add the list if its not empty
                set.add(aList);
            }
        }
        return set;
    }
    public static Map<Meal,List<Food>> getAllMealPlansMap(Set<Food> foods) {
        return foods.stream().collect(Collectors.groupingBy(f -> f.meal));
    }
    enum Meal {
        LUNCH, BREAKFAST
    }
    class Food {
        String name;
        Meal meal;
        public Food(String name, Meal meal) {
            super();
            this.name = name;
            this.meal = meal;
        }
        @Override
        public String toString() {
            return name + " " + meal;
        }
    }
    }