Cut calabaza in half. Scoop out seeds and stringy flesh. Discard stringy flesh, and reserve seeds for another use, if desired. Cut calabaza into 3” pieces; set aside.
Add 8 cups water to medium heavy pot over medium-high heat. Add brown sugar cane cones. Bring water to boil, stirring occasionally, until sugar dissolves. Stir in orange juice, orange zest and cinnamon sticks. Add reserved calabaza pieces to sugar mixture; return sugar mixture to boil. Reduce heat to medium low. Simmer, without stirring, until calabaza is completely tender and syrup seeps into flesh, about 30 minutes.
Using large slotted spoon, transfer cooked calabaza to serving tray. Cover with foil; set aside. Using slotted spoon, remove and discard cinnamon sticks and any loose calabaza skins from liquid.
Return liquid to boil over medium-high heat. Boil, stirring occasionally, until liquid reduces to thick syrup that coats the back of spoon (about 3 cups), 25-30 minutes.
Drizzle reserved calabaza with syrup; Serve calabaza warm, or at room temperature. Alternately, ladle reserved calabaza into small serving bowl, drizzle with syrup and serve with milk or ice cream.