Yes, it is possible for objects to move directionally towards "up, down, left, and right" without necessarily having a shape or magnetism. The concepts of direction and movement can exist independently of an object's shape or magnetic properties.
For example, consider a massless particle or a point object that does not possess any particular shape. Such objects can still exhibit directional movement. Similarly, the presence or absence of magnetism does not determine the ability to move in specific directions.
Directional movement is a fundamental concept in physics and can be described and studied irrespective of an object's specific properties like shape or magnetism.