As much as it bothers you, he's actually naturally doing exactly what he should be doing. I'll explain in a moment.
Changing voices for boys who sing is primarily difficult for two reasons: the physiological aspect and the psychological aspect. Because many books are written on this subject, I will very briefly address each citing information from a text I often employ (Becoming a Choral Music Teacher, Ward-Steinman).
You should listen to his voice approximately every six weeks to make note of changes in range, quality, and ease of production (pg.25)
Introduce songs that are comfortable for him to sing (in tandem with the higher pop songs):
"Ken Phillips, along with the other changing voice theorists, agrees
that vocalizing from the high voice downward is effective for
assisting with the registration problems experienced by adolescent
boys. In fact...a key to developing a high school tenor is to
vocalize the middle school boy[s] in the upper register." (pg.26)
So even though it's a bit annoying, vocalizing in the higher register is actually a productive way to: maintain a clear upper register after the voice change, assist with registration problems during the change, and maintain a positive attitude / atmosphere for singing during the change.
Right now, your son technically doesn't have a falsetto (which derives from old Italian meaning "false voice"). Oxford online dictionary defines "falsetto" as:
A method of voice production used by male singers, especially tenors,
to sing notes higher than their normal range
Since your son's voice hasn't changed yet, he doesn't yet have a "normal" range that he could then sing beyond, thus, he is simply using a high voice.
It's important for him to realize that most pop singers use what is called an aspirate (breathy) timbre. The breathiness comes from inefficient vocal cord resonance. For whatever reason, millions of people find it attractive (including your son). In addition, many pop stars sing with a "pinched" sound (lacking many overtones). You've mentioned in comments that you know how to help him open up his vowels. Proper vowel production is paramount to create a timbre rich in overtones. To you, because his voice hasn't changed yet, it will not have the resonance, power, volume, or overtones that a changed voice will, so he will continue to sound squeaky until his voice matures.
He should strive for a coordinated timbre in which the voice is open, relaxed, and the vocal folds are efficiently vibrating with minimum air loss.
Culturally, I would also make sure to expose him to a variety of positive male role models for vocalists. They shouldn't all be opera singers, but a healthy mix of cultures, musical styles, and styles of vocal production.
Since there are hundreds of books containing specific exercises, rather than put anything here, I'll just provide a couple of references for further reading:
- Ken Phillips - Teaching Kids to Sing
- James Jordan - The Choral Warm-Up