Pour crushed pineapple into a food processor; puree until smooth. Transfer to a large bowl.
Mix vanilla yogurt, half-and-half, cream of coconut, coconut vanilla Greek yogurt, heavy cream, sugar, and rum extract into the pureed pineapple. Refrigerate until thoroughly chilled, 30 to 60 minutes.
Freeze mixture in an ice cream maker according to manufacturer's directions, about 20 minutes. Transfer to a freezer container and freeze until firm, about 2 hours.