Heat a skillet over medium heat. Add squash and 6 tablespoons water; cover and cook 6 minutes. Uncover and stir in kale, oil, and salt; cook 2 minutes. Remove from heat.
Place 1/2 cup beans in a bowl; mash. Add remaining 1/2 cup beans and squash; toss. Divide mixture among tortillas; top evenly with cheese. Fold each in half over filling. Coat tortillas with cooking spray.
Wipe skillet clean; return to medium heat. Add 2 quesadillas; cook 2 minutes on each side. Repeat with remaining quesadillas.