I wouldn't include it. I doubt it makes much of a difference, but you yourself say it's not an appropriate item. I agree.
To expand a little bit, let me say this: the things you should put on your CV are those that directly connect to the job you want to have. Something like a job you had in an unrelated field is worthwhile in that it shows you were employed and not a hobo, but most other things that aren't really directly part of your career (jobs, education, publications, info about teaching, conference presentations, grants, etc.) is just a distraction, and will make it look like you are padding. I think high school achievements are especially bad in that they focus things too much in the past. Nothing on my CV goes earlier than my junior year of college (and probably I should cut that; nobody cares where I studied abroad).
Of course, it's good to have lots of things to list on your CV, but you also want to keep the average high. There are only a few things on there that are really important, and you don't want to distract people from what they are.
It would be great to discuss your experience as an Eagle Scout in a personal conversation, say if you have an informal dinner or drinks during an interview, but I just don't think the CV is the place for it.
EDIT: A point which a deleted answer raised also occurs to me: one reason to avoid putting extraneous things on your CV is that you can't control what associations people reading it might have. For example, the Boy Scouts of America have stirred up a lot of controversy with their stances on homosexuality, and for many people that maybe be the first thing that pops to mind. It's unfair to connect one scout to that, but people aren't logical. They read "Eagle Scout," they think homophobia, and they have a negative reaction to your application that they weren't even conscious of.
I recently read an application for graduate school by a student who listed on their CV membership in some political groups with which I vehemently disagree. I don't think that should affect my judgement of their file (luckily, I'm not on the committee, so I didn't have to try to make a judgement), but I honestly did not want to know about that aspect of their lives. I'm sure they thought it showed something about leadership, but to me it looked like very poor judgement.