Yes, children can get heel pain. Usually referred to as sever's disease- a condition developed from an injury to the growth plate. This can effect children with high arches or low arches. When the muscles in the calf are tight it causes the Achilles tendon to become tight causing strain on the growth plate.
If your child begins to sit out of normal activities, wants to be carried more often, or complain their feet hurt don't ignore them! To prevent serious foot problems from beginning in children or to prevent their current problem from getting worse, children should be seen by a doctor. You might hear these are "normal growing pains" but without the evaluation from a doctor, you don't know and could be ignoring a serious problem.