I can chime in about Boy Scouts although not at The Hill. Our son was a Life Scout when he entered BS as a ninth-grader and thought he would be able to continue along the path to Eagle Scout with a troop close to Choate that he contacted during the summer before entering. In reality, there was no time (at all) to participate in an off-campus activity, so he worked with his home troop leadership to put together aggressive summer and break plans to ensure that he could complete his Eagle requirements before turning 18. This meant that he was unusually heavily-immersed in Boy Scouts during his summers and during the long winter breaks. Last summer (between junior and senior years) was one nail-biter as he completed his project and board-of-review just one week prior to returning to BS and turning 18. If he did not pass his board of review before returning to school in the fall, he would not have made it.
My advice is to have your son meet with his Scoutmaster before leaving for BS to discuss in detail what he has left to do to achieve his Eagle rank and then map out a plan to ensure he can complete all requirements in the time he has allotted. This means that his troop leadership has to buy in to his unavoidable part-time participation in his troop and also ensure that he can serve in the required leadership positions that the Eagle rank requires. Do not count on partnering with a local troop.
As for college applications, spending summers in Boy Scout activities and achieving his Eagle rank will be among the best indicators of leadership potential as well as grit, determination, and endurance that your son can highlight to the colleges he applies to. Scouting is an extremely deep EC and if there is any way he can continue along the path and achieve that rank, it is well worth doing, first for his own deep sense of accomplishment, and second for what it says about his ability to go the distance. Our son feels that earning his Eagle is his most important life accomplishment thus far, even ahead of graduation from BS and entering one of his first-choice colleges. Once an Eagle, always an Eagle.