When my dd got intense and had strong reactions, I made her quit the activity because I said it was not good for our relationship. Eventually (age 6) she came to me with a proposal that I leave her alone to work it out and that she will not involve me in the process. Works great.
BUT.... she has never had those end of the world negative thoughts. Those are concerning (not the end of the world as I've known kids who are catastrophic thinkers) but still worth delving.
I'd say something like "that's an over reaction. Is that really how you feel? That if you can't do this one thing right, there's no point in doing anything else?" Process it with him. A LOT. And let him know that those thoughts can get out of hand. That they aren't true but if you keep saying them to yourself, you may start believing them.
If you can't get him to change his thinking pattern, I'd be seeking consult with a therapist.
ALso, try reading this book. It helped me tremendously with me glass half empty kid (who is not my super intense reactionary kid BTW)
http://www.amazon.com/The-Optimisti...-2&keywords=raising+the+optimistic+child
