Heat olive oil in a skillet over medium heat; cook and stir green onions until fragrant, about 1 minute. Add kabocha squash, shrimp, sugar, and salt; cook and stir until squash is lightly browned but still tender, 3 to 4 minutes. Remove skillet from heat.
Bring chicken broth to a boil in a pot; add squash and shrimp. Pour in enough water to cover squash by 1/2 inch. Simmer until squash is tender, 10 to 15 minutes.