Oh, you could hide such wonderful stuff in the tomato sauce, pureed carrots, sweet potatoes. Don't tell her though, do it while she is at school. Try 1 cup of the stuff with a jar of sauce. You honestly can't taste the stuff. I would probably let her drink more juice, it is better than nothing.
Getting them to help cook also sometimes helps. For my kids, if all we have is healthy food, it is what they eat. I would never force a kid to eat anything, but they tend to eat what is around them. Stop buying the junk, and you might see some change or wiliness to try other things.

When I had my daycare, I had lots of parents tell me their kids will only eat this or that. Well I was required to feed them a balanced diet, due to licensing rules, similar to the food programs school's use. They had no other choice but to eat a variety of foods, and you know what, the "picky" kids who actually had picky eater parents, ate all the stuff and liked it. They had no choice really, I never made a kid eat anything, but I would suggest that they try a bite at least. The kids would still say they don't like something (even if they at it at my home and like it) if a parent will give them something else. Their refusal is part of the control children can have over parents. You are trying, and that is better than a lot of other parents. It helps to have some good influences around them, kids who do eat the good stuff. and parent's who eat their veggies and fruit. Don't give in on the junk food, as long as that is in the house, they won't choose the healthy stuff. What does she eat in school?

As for the neighbor, tell her to mind her own business. You don't have to answer to her.