In a food processor, combine the flour, flaxseed, 4 eggs and oil, and pulse. Process until the flour is incorporated and evenly moistened. Lightly flour the countertop. Remove the dough and knead. Set the dough aside to rest for about 30 minutes.
Place the squash in the food processor. Add the cheese, 1 egg, salt, nutmeg and pepper. Process until well-mixed and smooth. Bring a large pot of water to a low boil. Roll out the ravioli dough into 2 long thin sheets that are approximately the same length and thickness. Place 1 tablespoon of the squash mixture every 2 to 3 inches down the center of one of the sheets of dough. Lightly brush the dough around the squash mixture with water. Place the second sheet of dough on top, and gently press to remove air and seal dough around each mound of squash mixture.
Cut the ravioli into squares or circles and place in the boiling water for about 8 minutes. Remove with a strainer or slotted spoon.