I agree with Whistle Pig's comments. It is true we have chicken and egg happening with this question/topic. For many reasons Eagles have a leg up versus non-Eagles.
navy2016 asked specifically about post-Eagle items like palms, assistant scoutmaster, and OA. With the various SA & ROTC applications, a minority of the applications have asked in yes/no format: Scout?, Life Scout?, Eagle Scout?, SPL?, JASM?, and ASM?. Never seen anything about palms or OA being asked although via a resume, essay, and remarks sections they could be included there if warranted and appropriate.
As said on these boards frequently, it's quality over quantity on ECs and leadership. Few Eagles get Palms and few Eagles become ASMs/leaders while in HS so those could help. However, if the admissions rep is familiar with Scouting they would know what level of effort is needed for a palm or to become an ASM and for ASM you need to fill out an application and get the Troop leader to sign it - that's it. It would be far more valuable to state what you have done AS an ASM rather than just state you are one.