A "False Dilemma" is also known as black and white thinking and kids with Aspergers frequently have this issue, especially when they are stressed out about something. To those who don't understand it, it can often be mistaken as pure, unbridled stubbornness. After all, you try to reason with them, provide perfectly valid and reasonable alternatives to the problem and they still insist on their point of view. It can be extremely frustrating for all involved.
This is a very interesting article on the subject, and while I'm not sure the suggested solutions would work, it does provide a great deal of insight on the problem. In this case, understanding the problem is definitely more than half the battle: False Dilemma: A Thinking Error