Okay, let's get this eating thing all squared away. Here's a nice little formula for you to use:
Your business: 1) What you purchase, serve, or otherwise make available. 2) Maintaining and modeling a healthy relationship with food.
Not your business: What your child chooses to eat.
Lots of parents have those reversed. They think that taking good care of their kids means that they must force them to eat certains kinds of food and restrict others. All kinds of goofy situations spring from that misconception - running the gamut from creating stress that impairs digestion, intractable power struggles, and on up to eating disorders.
When my kids were little, before I figured out this formula, I would serve them veggies for lunch and then sneak into the kitchen and break into my stash of chocolate while they ate. Then I realized that hypocrisy is at least as toxic to our bodies as junk food, and that the whole good food/bad food dichotomy was not healthy either. It set food up as a force that was stronger than they were, and that's not an empowered position to make choices from.
I also wanted my kids to maintain awareness of their hunger and sensory feedback, and placing strong taboos on certain foods only thwarted their own inner guidance. Besides, if children learn what they live, it seemed like a good idea to bring my own behavior into alignment with what I wanted them to become. So I stopped prohibiting and demonizing certain foods, and focused instead on my own integrity and listening to my own body's signals.
Assuming your child is of sound mind and body, he or she truly does not want to starve. Hold firm to your values by preparing foods you believe are nourishing and eating them yourself. Don't bother bribing with dessert - serve fruit and make it part of the meal.
Talk about nutrition, help your child get involved in the shopping by teaching her how to read labels, letting him pick out which veggie to eat with dinner, etc., and then relax and enjoy your meal together. If you are packing a school lunch, put a little note in there, or something that expresses love and appreciation.
Many times, when kids feel harrassed about food, they simply decide it's not worth arguing about and go underground; pitching out that nice hummus sandwich and eating cupcakes and twinkies from their friends. It's okay. It happens. We cannot control what they do with what we pack.
Kids need to learn how to listen to their own bodily signals, and we do them no favor when we try to protect them from experiences that are natural feedback loops. They need to FEEL the sugar crash before they can decide they prefer to avoid it. Just keep preparing and packing the nutritious stuff and don't worry about it. They will carry out their own experiments.
You might also consider collaborative problem solving. Express your values and concerns about nutrition, and your wish to provide nourishing options. Then find out why your child is not eating her lunch, and put your heads together to come up with some alternatives that work for both of you. You'd be amazed at the variety of food options available these days! They even stock vegan gummy bears sweetened with fruit juice at my local Vitamin Cottage. There are many win-wins to be found.
Ultimately, love and respect are the most nourishing things you can serve to your kids. Kids eat that right up every time, and you don't even have to go farther than your own heart to procure an abundant supply.