My niece did the same thing. After trying not to make a big deal out of it, it came out a little later (and after a few MONTHS of this happening) that she thought she was fat, and didn't want to eat because of that. FAT is about the farthest thing from the truth with this girl, she is a bean pole, but in her group of three other girls, two are on the chunky side, so it may have been in solidarity with those two, not that that makes it right either. What's funny is she would come to my house after school and literally scarf everything in sight, she was so hungry! It was pretty much just a phase, as she is eating again at school and I don't think anything the school, her mom or I told her that made a dent in her thinking; I think she just got tired of being hungry.
Is it possibly the food is just not that appealing? Try home lunches, or if that is a no-go for whatever reason, pack a snack in her school bag; a breakfast bar, raisins, a banana, or something, and explain that even if she is not hungry for lunch, it is important for her brain to get food at lunchtime so she can continue to learn in the afternoon.
My own daughter is not a breakfast person, and would really prefer NOT to eat in the morning, but after waging war over it--you have to have SOMETHING--the fuel for the brain conversation worked, and she now grabs herself something small every morning as part of her routine, usually half a banana and milk, or a granola bar and a few sips of OJ.
Good luck!