My 8 yo grandson says similar self-loathing things when he's feeling sad. He's in a special school and receives all sorts of emotional support. His teacher is not concerned about it and we've learned that the best way to deal with this is to be sympathetic. We say, we know you're feeling bad but we want you to know that you are a good boy. (He cries and says he's a bad boy.) I hold him and let him cry.
This usually happens just before bed when he's tired. He will usually stop crying and fall asleep in 15-20 minutes. He does do this at school from time to time and they give him some emotional support. He can have some time to himself with an aide.
I think that one reason my grandson feels this way is that someone in his life is telling him he's a bad boy, perhaps not by using those words but by treating him as a bad boy. Perhaps by withdrawing love when he's not behaving appropriately. I know he gets spanked which is shaming. This creates self loathing instead of helping him to be confident in his ability to learn.
My grandson doesn't say he wants to die tho his demeanor seems to indicate that possibility. For me it indicates that he needs support. He needs to be told regularly that he's a good boy: be praised when he's doing well. I tell him often that I love him and that I'm so very glad he's in my life.
As to what to suggest at the school conference, I'd go in believing that this is bullying and discuss ways to stop it. This is not a new situation for the school. Schools, now a days, have plans in place for handling bullying. I would also ask that my son have time with the school counselor so that he can learn how to be more assertive and not wait until he snaps. I suggest that once he's more confident he'll be less of a victim.
When you say he snaps, I wonder if you're saying that he eventually responds by also throwing rocks. Frequently bullying creates more bullying going in both directions. When one defends himself in anger the result is more bullying behavior. Throwing a rock back creates more anger and a vicious cycle. Children need help to learn how to react to bullying so that they're not responding in anger that creates more serious behavior. My granddaughter's school has a teacher that teaches the children how to communicate so that there is less bullying. The teacher directly teaches ways to get along. My granddaughter is also 11.
When she was younger she was having difficulty with some bullies and she did react by hitting back. She was then given the opportunity to be in a Friendship Group in which they were taught how to handle these sorts of situations. The group also worked on building self-esteem.
Modified: I suggest that when your son is this upset his self esteem has hit bottom. It may be that he thinks he should not react by hitting back, etc. He's failed in his eyes. I suggest that the way to counter this is to tell him it's OK to get so angry but instead of doing this try walking away and telling a teacher how angry you are because those boys called me names or whatever happened. Emphasize that the feelings are OK. It's the action that you're wanting to help him change.