Feed squash through an electric juicer. Pour juice through a fine sieve into a small bowl; discard solids. You will need 2 cups juice.
Fill a medium bowl with 4 cups cold water; submerge gelatin sheets and let stand until softened, about 5 minutes.
In a medium saucepan, combine juice, Parmesan, and butter. Squeeze excess water from softened gelatin sheets; add to saucepan, whisking to incorporate. Season with salt and pepper.
Pour mixture into a 12 1/4-by-9-inch rimmed baking sheet; transfer to a refrigerator. Refrigerate until set, about 2 hours.
Just before serving, bring a large pot of water to a boil over high heat; add wonton wrappers for 10 seconds. Using a slotted spoon, remove wrappers, and lay each flat on work surface.
Remove baking sheet from refrigerator; cut gelatin into 1/2-inch cubes. Place a cube on top of a wrapper, folding like an envelope to enclose; repeat with remaining wrappers. Transfer ravioli to a large microwave-safe dish; place dish in microwave on high until gelatin melts, about 5 seconds.
For each serving, place a ravioli in each of four large soup spoons; top with truffles, and garnish with chives. Repeat with remaining ingredients. Serve immediately.