Place squash cubes in a large pot and cover with water; bring to a boil. Reduce heat and simmer until tender, about 20 minutes. Drain squash. Mash and set aside to cool.
Preheat the oven to 400 degrees F (200 degrees C).
Heat a dry skillet over medium heat. Add chicken and mushrooms; cook and stir until chicken is no longer pink, 5 to 10 minutes.
Drain chicken and mushrooms. Add mushroom soup. Fill empty can with water and pour in. Cook until mixed well, 2 to 3 minutes.
Add flour and 1 egg to the cooled mashed squash. Mix to form the pastry. Transfer to a well floured surface. Divide into 6 pieces. Roll out 3 pieces to fit 3 nonstick pie pans. Prick the bottoms with a fork.
Spoon chicken and mushroom filling over the pastry. Roll out remaining pastry and place over the filling. Beat remaining egg and brush over the tops.
Bake in the preheated oven until golden, about 30 minutes.