Ok, taking a stab at this... My DD used to be terrified to eat in school, she does eat now but it has to be a packed lunch that she watches me make. She has an anxiety disorder that has led to a condition called "emetophobia" which is an extreme fear of throwing up. She is absolutely TERRIFIED she will get sick and throw up so she's very careful about eating. I'm not saying this is your DD's problem, but maybe it is an anxiety related thing? I learned after dealing with my DD that children are developing anxiety disorders at very young ages and they can manifest in many ways and there's not always a clear reason of what is stressing them out. We still don't know what triggered this in our DD other than possibly genetics since I've had OCD and Anxiety Disorder since a young age as did my mother.
Anyway, another thing- I NEVER would eat lunch at school either, but that was when I was in high school and had a bout of anorexia. I know you're DD is young but is she ever mentioning fears of getting fat? When we took our DD to a therapist we also learned that children are becoming more and more aware of "society's expectations" that girls should be thin to be pretty.
I don't really know the whole story with what's going on with your DD so I hope I haven't overstepped myself.