push assembly top gif

this gif demonstrates the push assembly motion as well as the core assembly underneath.
the push assembly transfers motion to the opposite side core mechanic assembly piece,
in effect 'pulling' the opposite side, thus a coordinated push of all four simultaneously is required.
in this gif you can see the push assembly range of 15 units, a pause, the push assembly is rendered transparent, and then the core assembly completes its range of motion.
in other words, the push assembly 'pops out' the core assembly buttons half way, and then the remaining motion must be completed by pulling the core assembly buttons

with box sides

here you can see the box sides.  each side will have two square slots. central slot will house core mechanic pull button and right side slot houses push button
note that bottom push assembly is hidden in this rendering

push assembly top

the push assemblies will lie both above and below the core mechanic pull assembly.
each push assembly consists of two identical pieces that transfer the push motion from the square button on the side to the central pull button on the opposite side
the core mechanic's pull buttons will sit flush with the outside of the box when in the fully closed position. the push assemblies allow the player to 'pop out' the central assembly's pull buttons. 
the push assemblies' range of motion is only half that of the core mechanic's full range of motion
full range of motion = 30 units
push range = 15 units

core mechanic assembly

fabrication technique will utilize layering to avoid end grain of wood facing out 'button'
consistent thickness of 10 units for all pieces
inventory.box_pieces:
    2x core_align dim=10x20x30
    1x core_button dim=10x30x30

x4 core_assemblies total:
    8x core_align
    4x core_button

core mechanic gif

The core mechanic assembly

4 interlocking pieces that must be pushed / pulled together: