I wish to begin by saying that this is certainly not a situation I would ever wish to be in. However, if I were faced with this situation or one similar to it in which one of my loved ones were lying in a hospital bed with a machine breathing for them with no hope of improvement, I would pull the plug. Also, if I were ever the one lying in that bed in a persistent vegetative state or in a state of constant, irrelievable pain like this little boy, I'd want someone to pull the plug. The real shocker here is that for all intents and purposes, I am considered ultra-right wing (Reformed Presbyterian) with the possible exception of my views on situations such as this. My thoughts in this sort of situation are a bit more pragmatic and utilitarian than most other Christians.
1) The child is in pain. Keeping him on the respirator so that he remains alive only prolongs that pain. It is our duty as good and moral people to avoid inflicting undue pain on others. Now, one of 2 things happen in this situation, depending on your worldview: A) If there is no God and no afterlife, then when the child dies he simply ceases to exist, thus ending his pain and anything else that he feels. B) (My particular worldview) If the child was baptized into the Covenant (because his parents "vouched" for him, it's sort of complicated to explain it), then his soul is given a heavenly body with the rest of the saints. On the other hand, if he is not the child of believers, then we get into a grey area (there is much debate within the church about whether young children who have not had the opportunity for salvation are bound for heaven or hell, there is simply little evidence in the Bible for either one).
2) There are others who need that bed worse. I realize that it sounds cruel, but let's assume for a minute that this hospital is filled to capacity. Somebody else can't get that bed until he dies or gets better (which in the case of an incurable degenerative disease that has progressed that far, it's not going to happen). If there is someone whose life can be saved if they get that bed within the next 24 hours, then the plug should be pulled immediately.
3) As for why a good God would allow these sorts of things to happen, my only answer is that God knows His plan and I don't. That is the simplest way I can put it, and I'll admit that it sounds like a total cop out. I suppose that is where faith comes in, even though that is also a cliche, Sunday school answer.