I cannot stress this enough. Not everbody loves plain flavorless vegetables. And here is where primal and mediterranean cuisine can do miracles. The fact that there are numerous dishes that are exclusively made with vegetables, cooked in olive oil/ good fast and herbs and served with feta is an effective reason. Instead of handing some carrot sticks to your kids, try making the vegetables almost decadent by cooking them.
One average serving for a child, will provide between 3-4 servings of vegetables. So kids actually do not need to worry about getting more vegetables the rest of the day.