In an egg drop challenge, Scouts try to design a container which can prevent an egg from breaking when dropped. This is an excellent project to get kids interested in science and engineering.
Rube Goldberg was a cartoonist and engineer who drew elaborate contraptions to accomplish simple tasks. This article from TinkerLab gives step-by-step details for creating a Rube Goldberg machine.
BSA has a Cub Scout pack meeting plan called Heroes in History related to citizenship. The Heroes in History meeting plan features games, group activities, songs, and more to help Cub Scouts learn the value of honest leadership.