In a large, heavy pot, stir the water, brown sugar, and cinnamon sticks together until the sugar dissolves. Add the butternut squash and bring to a boil over high heat. Reduce the heat to low and simmer, uncovered, until the liquid is reduced and syrupy, and the squash is very tender, about 2 1/2 hours.
Place the reserved seeds in a colander and rinse to remove any pulp. Dry thoroughly on paper towels. Heat an 8-inch skillet over medium-high heat. Add the squash seeds and cook, stirring frequently, until lightly toasted. Set aside to cool completely before using.
Remove the cinnamon sticks from the squash and discard. Let the squash cool slightly and serve warm or at room temperature with the toasted seeds.