Remove giblets from squabs; set aside. Combine squabs, celery salt, and paprika in a large Dutch oven; add water to cover. Simmer 1 hour or until squabs are tender. Add reserved giblets during last 20 minutes of cooking time.
Remove squabs and giblets from broth; reserve 2 cups broth in Dutch oven. Chop giblets. Remove meat from bones, and chop. Spoon half of chopped squabs into a greased 2-quart casserole. Spread half of chopped giblets, mushrooms, and eggs over top. Sprinkle with 1/4 teaspoon pepper. Repeat procedure with remaining squab, giblets, mushrooms, eggs, and 1/4 teaspoon pepper.
Combine flour and milk; stir until smooth. Pour flour mixture into reserved 2 cups broth; cook, stirring constantly, until thickened and bubbly. Stir in remaining pepper and salt. Pour gravy over layered mixture in casserole.
Prepare pastry. Roll dough to 1/2-inch thickness; place over pie leaving a 1/2-inch overhang at edge of dish. Turn pastry edges under; press firmly to rim of casserole to seal, and flute. Cut slits in top for steam to escape. Bake at 400° for 25 minutes or until crust is lightly browned.