Preheat oven to 400 degrees F (200 degrees C). Line 12 muffin cups with paper liners.
Mix flour, sugar, flax seed, baking powder, baking soda, and salt in a bowl. Beat eggs, milk, applesauce, lemon juice and canola oil in another bowl until smooth. Stir flour mixture into egg mixture until just moistened; fold in cranberries. Fill prepared muffin cups about 2/3 full.
Bake in the preheated oven until a toothpick inserted into the center of a muffin comes out clean, 18 to 20 minutes. Cool in the pans for 10 minutes before removing to cool completely on a wire rack.