The first thing to do is stop worrying. Children won't starve themselves, and they will eat when they're hungry, as long as we don't interfere with their natural hunger signal. When allowed to choose for themselves, they will instinctively take what's good for them.
To coerce children into eating when they're not hungry is counterproductive. It's teaching them to eat for the wrong reason. The only reason we should eat is because our body needs fuel. Food is neither a reward, nor a punishment. Fruit and vegetables are part of a normal diet, and eating them should not be a chore that will be rewarded with sugary dessert.
If your children refuse to eat vegetables, start cleaning out your fridge and cupboards. Sugary drinks are a big no-no, and need to be replaced with plain water. Get rid of lollies, biscuits, chocolate, chips and so on, and stock up on celeri, carrots, cucumber, tomatoes, apples, pears, oranges, bananas, kiwi fruit, etc. If they're hungry, offer them fruit. They might protest to begin with, and even have tantrums. Tell them there's nothing else in the house, and you can't go shopping right now. Give them strawberries. Every kid I know loves strawberries. You can even top them with yoghourt, to make them more appealing.
When you cook, try to incorporate finely chopped or pureed vegetables into your spaghetti sauce, as they won't even notice them. Lasagne and pasta bake is also a great to conceal all kinds of greens. Try offering raw vegetables to your children, instead of cooking them. My kids have a passion for raw carrots, and will chew on them while watching their favourite TV programmes.
If your kids still refuse all fruits and vegetables, no matter what you do, don't panic. It might be just a phase, and they will grow out of it eventually, especially if you gently persist, without being pushy. Give them a good vitamin and mineral supplement. A fibre enriched fruit juice is also a good idea, to prevent constipation and other problems caused by lack of roughage. Don't force them to eat what they don't want. Instead, allow them to be themselves and in control of their decisions. Be flexible with mealtimes and food choices. It's not harmful to have chocolate or ice cream from time to time. Anything in moderation is okay.
Be a role model to your children. They will emulate your behaviour, and if you eat healty food, they will follow your example eventually. As they become teenagers, their taste will change and they will also be more open to foods that they've never tried before.
Congratulate yourself for being a parent and remember that you're doing the most important job in the world.