Bring chicken stock, kelp, and dried anchovies to a boil in a large pot. Cover and cook over medium heat, about 10 minutes. Strain broth, discarding kelp and anchovies.
Cook pork belly in a large pot over medium heat until browned, 3 to 5 minutes. Transfer to a small bowl; reserve drippings in the pot. Add red pepper flakes; cook and stir until sizzling, about 30 seconds. Return pork belly to the pot. Add strained broth, zucchini, onion, kimchi, serrano chile peppers, ginger, fish sauce, and sugar. Cover and simmer soup until flavors combine, about 15 minutes.
Stir tofu, shrimp, mussels, clams, mushrooms, and garlic into soup. Simmer until mussels and clams have opened, about 5 minutes.
Break tofu gently into chunks. Ladle boiling soup into serving bowls and crack an egg into each. Garnish with green onions.