After having bacon and eggs for breakfast, quite often I send lunches that are uneaten because the ten year old says she isn't hungry. Should I push her to eat during the day or just let her skip lunch?
Meh, not harmful in my opinion. If they are hungry enough they'll eat it. With overweight and obese children as the new model for normal size I'm perfectly content to let my kids skip a meal here and there. In fact if they don't like whats for dinner they get to skip that too :)...
EDIT: Should add that my kids definitely tend to double down on breakfast or dinner if they skip a meal. I feel my kids satiety mechanisms haven't been all jacked up yet and find their intuitive eating and not eating for that matter something I like to emulate.
Do you think she truly isn't hungry, or does she perhaps not like the lunches you make - is she a picky eater generally? Or is she conscious of her lunch not "fitting in" with the other kids' food (I definitely experienced this as a kid)? Is she possibly buying junkie cafeteria food instead (I definitely did this as a teen)? Is she a good eater at home? Is she underweight for her age, any health problems? Or do you honestly believe that at lunchtime all the other students unpack their lunches, and she's just sitting there watching them because she's legitimately not hungry. And what kind of lunches are you packing for her?
I'm not trying to sound accusatory, just want to get a full picture of things. Devil is in the details.
I had to admit your post raised a red flag for me, because my teenage adventure into EDs began with skipping lunch. Although I did hide it from my mom - I just threw them out and told her I ate them. And I know girls are starting EDs earlier and earlier these days.
Is she super body conscious? You said in an earlier comment "she's currently in the toning up stages." Just something to be on the alert for. As if you need more to worry about with a pre-teen...
My kids both do this, because they want to play, and recesses are short. I've talked to them about it, and I've asked them to make a point of eating the protein item first, so if they don't have time to finish, at least they'll have something nutritious in their belly. And I give them big breakfasts and big snacks (which I conceptualize as a late lunch).
I'd trust a kid's judgement of hunger over my own, but there are other factors to consider as has been mentioned. There's no reason she should be hungry at lunchtime, so if she seems hungry enough at breakfast and dinner and is eating well and a healthy weight etc. then I wouldn't push her. Perhaps find something that she can take in case she's hungry but which will keep like a bag of nuts.