Toss shrimp with ginger in a bowl and marinate, chilled, 30 minutes.
Cook shallot, garlic, and anise in butter in a 3-quart heavy saucepan over moderate heat, stirring until shallot is softened, about 5 minutes.
Add squash, stock, and water and simmer uncovered until squash is very tender, about 20 minutes.
Remove star anise.
Puree soup in 2 batches in a blender until smooth, about1 minute per batch, then transfer to cleaned pan and keep warm, covered.
Sprinkle shrimp with salt.
Heat oil in a large skillet over moderately high heat until hot but not smoking, then sauté shrimp in 2 batches, stirring until just cooked through, about 3 minutes per batch, transferring to paper towels.
Bring soup to a simmer and season with salt and pepper.
Divide among 8 bowls and mound 3 shrimp in each bowl.