Using a food processor fitted with a shredding disk or a hand grater, coarsely shred enough squash to measure 2 cups. In a 4- to 5-quart slow cooker combine the 2 cups shredded squash and the next six ingredients (through pepper).
Cover and cook on high for 2 hours, stirring once halfway through cooking. Stir in semisoft cheese; top with American cheese. Cover and cook for 5 minutes more; stir gently to combine. If desired, stir in enough milk to reach desired consistency. If desired, top with parsley.