Cut steak diagonally across grain into 1/4-inch-thick slices. Combine steak and next 4 ingredients in a large zip-top plastic bag; seal bag, and chill at least 1 hour. Remove steak from bag; discard marinade.
Combine 1/4 cup soy sauce and next 5 ingredients; set aside.
Cook rice according to package directions, omitting salt and fat; drain.
Lightly beat eggs and egg whites in a medium bowl. Dredge steak strips in flour; dip in beaten egg (do not return steak to flour).
Heat 1 tablespoon oil in a large nonstick skillet. Add steak, in batches, and cook 2 minutes on each side or until golden, adding 1 teaspoon additional oil to pan for each batch. Serve steak with sauce and rice.