Combine first 3 ingredients in a large bowl; cover and chill 30 minutes.
Preheat oven to 350°.
Heat olive oil in a medium skillet over medium-high heat. Add onion; sauté 3 minutes. Add onion, feta and Parmesan cheeses, chopped mint, and black pepper to squash mixture, stirring to combine.
Working with 1 phyllo sheet at a time (cover remaining dough to prevent drying), place 1 sheet into a 10-inch deep-dish pie plate coated with cooking spray. Gently press sheet into pie plate, allowing ends to extend over edges of pan; lightly coat with cooking spray. Place another phyllo sheet across first sheet to form a crisscross design; lightly coat phyllo with cooking spray. Repeat procedure with 2 phyllo sheets and cooking spray.
Spoon squash mixture over phyllo. Place another phyllo sheet over squash, repeating crisscross design with remaining phyllo and cooking spray. Fold in edges of phyllo to fit dish and form a rim. Bake at 350° for 40 minutes or until golden brown.