There are 2 common systems used to retain shelf support pegs in a refrigerator wall. The most recent design has a center plastic push-pin that when inserted into the support causes barbs on the support's base (see illustration) to expand and lock the peg into place in a square hole in the cabinet wall.

This type can often be identified by looking for evidence of a (now filled) center hollow. To remove these the center push-pin has to be pushed right through the support peg using an awl or similar tool, to release the barbs. After separated, the push pin may be able to be retrieved from the cabinet insulation to be reused if necessary.

The other type of shelf support commonly used is the 'Simmons' fastener. This support design is inserted into its mounting hole in the cabinet wall and then twisted to lock it into place or to remove it. Blunt nosed pliers work well for this when held perpendicular to the refrigerator interior wall surface.

Sometimes you'll get lucky and a 3/4" open end wrench will fit the base of the shelf support which then can be used to rotate the support out of, or in to, its mounting hole.

Once grasped firmly by the support base, the peg needs to be rotated 90° clockwise to release the support peg's retaining wires. After this is done, the peg should be loose and able to be maneuvered. The loose support may have to be manipulated slightly to align the retaining wires through the 'keyhole' mounting slot in the refrigerator wall. Once this is done, the old support can be removed.

To install the replacement (or reinstall the original), insert the part in the 'keyhole' shaped hole and again holding the base securely, rotate the peg 90°  clockwise . It should now be tightly held in place. Be sure to take into account the rotation when initially inserting it into the hole so it will not be upside down when tight. If it is upside down, continue to rotate the support clockwise to remove it again and start over.