There is just no way,
Melamine is a "chipped board" with a thin plastic coating on each side.
1/4) would be 2 thin to make the connections. From what I read even the thick Melamine "loosens" up after a while.
If you really want to make it out of wood, I would use a good grade plywood, screw and glue the corners.
The frame is the basis of the whole printer,, this would be like building a house on a subpar foundation.
Nick