Boundary labeling connects each point site in a rectangular map to a label on the sides of the map by a leader, which may be a straight-line segment or a polyline. In the conventional setting, the labels along a side of the map form a single stack of labels in which labels are placed consecutively one by one in a sequence, and the two end sides of a label stack must respect the sides of the map. However, such a setting may be in conflict with generation of a better boundary labeling, measured by the total leader length or the number of bends of leaders. As a result, this paper relaxes this setting to propose the boundary labeling with flexible label positions, in which labels are allowed to be placed at any non-overlapping location along the sides of the map so that they do not necessarily form only one single stack, and the two end sides of label stacks do not need to respect the sides of the map. In this scenario, we investigate the total leader length minimization problem and the total bend minimization problem under several variants, which are parameterized by the number of sides to which labels are attached, their label size, port types, and leader types. It turns out that almost all of the total leader length minimization problems using nonuniform-size labels are NP-complete except for one case, while the others can be solved in polynomial time.