Am I being a "mean" instructor, denying an extension on a take home exam
You give a lot more information about the student than seems relevant to the question.  That the student is applying to MIT for instance is neither here nor there.  

That your student is in high school may be slightly relevant.  It feels awkward to have a high school student in a college/university course when they are not doing well, because usually students (especially in mathematics) take such courses because they are too advanced or gifted for their high school programs, but that does not automatically make them prepared for college/university standards.  If you are an inexperienced instructor (and I think you are: more on that later) then it is a good idea to talk to colleagues to get some perspective on how high school students are typically dealt with.  This may well be different at a community college than at a research university, by the way.  I will say that my attitude in dealing with high school students in (research) university math courses is: if there are any special rules or dispensations that apply to them, these should be made clear at the time of their enrollment in the course.  In the absence of such rules, I try to treat high school students identically to all other students (keeping in mind that "all other students" is a large and inhomogeneous group).  I would certainly avoid talking to them in a way that would not be appropriate or respectful for an adult university student.  E.g. you write


  I think he lacks maturity and I told him if he wants to go to MIT that he should act like an MIT student and quit complaining and get his work done.


This sounds rather condescending to me.  As I said above, that the student wants to go to MIT is really not relevant to your course, and dragging that into the conversation is unhelpful.  By the way: telling someone to "act like an MIT student" presumes some involvement / affiliation with MIT on your part.  If you don't have that, what you say can come off as obnoxious.  I have taken classes at MIT (as a Harvard PhD student) and have some colleagues on the faculty there (for instance, I spoke there last month).  But I would not tell anyone to "act like an MIT student": while MIT students are in the aggregate very impressive, they are people, not stereotypes, and in my experience evince a wide range of behavior, including complaining.

Coming back to the matter at hand: honestly, what strikes me most is the way you are teaching your course.  Giving an exam which is too hard -- even much too hard -- is something that instructors do often, and it is not necessarily a problem but needs to be dealt with very carefully.  Turning an in-class exam into a take-home exam after it has been taken in class seems like a prohibitively poor idea to me.  If you are not sure about why, you may want to ask that as a separate question, since it would be helpful to get a range of responses about this.  By the way, I also find


  "give thanks, the exam is take-home now and due Wednesday". 


to be a bit obnoxious.  A week after an exam was given, a student is expecting to receive their graded exam back again.  Coming into class on a Monday and learning that they are getting a "pop take-home exam" is something that a wide range of college/university students would not be thankful for.  Most students are very busy with a range of activities (academic, extracurricular, work, health, family...), and very busy people plan their schedules in advance.  Giving the exam back without a grade, asking the students to do it by the next class, and then forgetting to communicate that to students who missed that one class, is not great behavior on your part: honestly, it looks a bit lazy to me. 
You say that you want to post the solutions in time to have the students study for the final.  That's good.  However, remember that you were the one who created the time crunch in the first place by making an exam you were so unhappy with so as to mess with it later.

Okay, what do I recommend?


You won't want to hear this, but: giving a take-home exam in an undergraduate math class in 2016 is probably a bad idea full-stop.  As a participant on math.stackexchange, you know well that your students can get solutions to even a way-too-difficult undergraduate differential equations class rapidly without leaving home.  I have colleagues at SLACs who feel confident that the majority of the students take the honor code seriously enough not to do this and feel that the cheating by a small minority of students is more than offset by the gain in learning from students who work through a carefully crafted take home exam.  When I was a postdoc, I had the experience of students cheating on a take-home exam (at a research university) and saw the poisonous effect their higher grades had on the morale of the other students, so for me even a small minority of cheaters is a high price to pay.  But this is not a carefully crafted take home exam: it's a too-hard in class exam.  And moreover it's a take home exam that was already given as an in-class exam.  Don't you think that many of the students spoke to each other about the exam afterwards?  And that a few of the students may have been interested enough to look up the answers online and/or talk to outsiders (e.g. more advanced students, tutors, instructors) about it?  I see no way to guarantee academic integrity in this situation.  
You could make a different exam for your student and all other students who want to take it / can't solve the take home according to the deadline(s) that you rather precipitously announced.  Alternately, you could admit that the assessment coming from this midterm is not the greatest and just offer to drop everyone's lowest midterm grade.  In fact I usually drop the lowest midterm grade in my undergraduate classes conditionally on satisfactory performance on the final.  (And in fact I often don't announce it until near the end of the course, because I want all the students to take all the midterms.) I find that students are very happy with this, and since I too often give exams that are a bit more difficult than I had wanted, this ends up with course grades in the range that I want.