Like Popeye, Gilligan is (1) a sailor who (2) gains super-strength from (3) eating spinach -- only this time it's radioactive spinach.
The voice on the radio that announces the loss of the radioactive seeds is Whit Bissell - uncredited.
Mary Ann and Gilligan discuss carrots and their ability to improve eyesight. This was WWII propaganda intended to hide the Allied use of radar.
Carrots are Mary Ann's favorite vegetable, sugar beets are Mrs. Howell's favorite vegetable, and spinach is Gilligan's favorite vegetable.