Boys State vs. Eagle Scout
What we do know about Boys State & Eagle Scout is that USMA widely publicizes the numbers of cadets in each entering class with either honor. Which means that any prospective cadet should seriously consider pursuing one or the other, or both, since this is one of the few really clear clues USMA provides about what they are looking for in a successful candidate.
But they are obviously quite different. Eagle Scout takes a lot longer to earn. It rewards focus, discipline, work ethic, and perseverance, along with civic mindedness, patriotism, etc... But almost any scout with a sufficient degree of these qualities and some support from their family and their troop can become an Eagle Scout. A really strong troop can produce multiple Eagle Scouts each year. That doesn't mean it is easy.
The American Legion sends students to Boys State who have demonstrated leadership qualities, civic mindedness, patriotism, academic prowess, etc... These are many of the same qualities exhibited by Eagle Scouts. The distinction with Boys State is that the American Legion generally selects just one student from each high school to attend Boys State. Thus, a group of military veterans reviews the applications/nominations of the top candidates at a given high school, interviews them, and chooses just one student to attend Boys State. It can vary quite a bit depending upon the school, but at many high schools the competition can be quite fierce. In competitive schools, the Boys State delegate will have displayed the same focus, discipline, perseverance, as the Eagle Scout does in maintaining his grades, playing a varsity sport, and/or participating in other activities on campus.
For the non-scout, Boys State provides a terrific opportunity to enhance their USMA application. My son was a Cub Scout, but stopped scouting in middle school when he became more interested in sports. Thus, he wasn't going to be an Eagle Scout. But he worked hard in his sport, and school, and student government, and was able to get to Boys State, which seems to carry similar weight with USMA. But it isn't a sure thing as there were many very strong candidates competing for just one Boys State slot and he was very fortunate to be able to attend Boys State.
A side note - My son attended SLS and California Boys State almost back-to-back last summer. It was remarkable how many of the same young men he saw at both events.