Should I trust this kid?

It would be inappropriate to install this on your computers, I think. No amount of trust in the student would allow for that, in my opinion.

Even if the code is kosher, you would have to compile and distribute it yourself to ensure the actual app is legit, and then do the same for every update to avoid him sneaking something in later. It's not feasible to do this responsibly.

I agree with @alexjdale. The second you deploy it to machines, you are responsible for supporting it, so even if is 100% clear, you are now responsible for the upkeep and/or fallout something like this might cause.

I agree that this is not a good idea. But please give the kid a project that would be useful to you so he can learn and see some success!

JAMF would not exist with you guys having been around for its early beginnings. :) At least I thought I remembered hearing about how one of the JAMF guys started managing computers in high school and wound up going back and selling Casper Suite to the School district afterward. I concur with @Josh.Smith get a contract in place and employ the young mans talents some sort of extra credit if he can maintain it and make it more helpful as the year goes on or create other tools as well. Trust and guidance could help this guy be the next great computer genius. Of course that is easy for me to say :P

Coming from a student perspective. Finishing High School last year, this is something I wouldn't trust. All I can say is maybe let him distribute it around to people, sure, but keep an eye on him, as soon as that becomes malicious, that's on you. On him of course as well but you letting it happen.

Make sure to clear this with your boss so in the event this comes back, it doesn't bite you in the arse.

I agree with @Josh.Smith and @nessts. This is a good opportunity to mentor if you've got the bandwidth.

You've linked to the student's compiled app, which makes it difficult to review the code. Explain the concept of open source software to the student and then explain how to post his work on Github for his peers and yours (us) to review. Github requires its users be at least 13.

The idea the student's project could reach beyond his own school might be appealing to him and might give him some good experience receiving critical feedback. And feedback sometimes includes praise, which may be something he needs.

As a former educator I agree with @talkingmoose, @Josh.Smith, and @nessts. Mentoring is the way to go, however if it's something you can't take on, have a conversation with the teacher about his skill set, the tools that he's got available to him at school and home (maybe there are none at home?), but most importantly, about his potential. From past experience, I had a student quite like this, and he was perceived as a trouble maker only because he was seeking attention in class, due to personal issues at home. The teacher didn't trust him because they didn't understand the technology. I intervened and had a frank discussion with the student, at the teacher's request, about ways he could help the teacher with the tech in the class. Things improved and the teacher's trust grew for the student. Granted, I know not all cases are like that, but in that case, it worked.

Turned out, the kid just wanted to succeed at something he was good at/interested in.

As for as the under 13 issue, if he is, have his teacher/school counselor/principal reach out to the high school or community college to see if there are established classes or clubs he can visit and maybe even work with.

You may want to pull that code down and sanitise it before posting again, it has the student's name in the code.

I've had a brief look and it all looks fairly innocuous, but you would want someone with better Obj-C knowledge than me to confirm this before making any decision.

My generally feeling though is with many of the above, this kid deserves a chance and is probably disruptive because he needs some help/mentoring/trust put into him. This could well be his opportunity to get headed onto a constructive road rather than the destructive cycle that many of these bright but misunderstood kids get into.

If you can help him, do so.

Thanks all.
Good advice.
I've taken down the link until it can be sanitized, and I'm going to run it by a couple coder friends, because I really want to encourage him if it is indeed safe. Going through the code it seems pretty innocuous, and I want to be able to at least allow other students or his friends to use it if it works well. Our students can install applications for themselves, so there isn't really anything I can do if he wanted to install it on friends machine's anyway.
We'll be working with him more this year, but he is 17 so this is his last year.
Thanks