"This code won't (probably) work with rectangular BBOX! You have to use square BBOX"
The code theoretically works with all kinds of bboxes, square or not.

The problem is, as already identified, the OBB collision system. Because of that you can move the player's hull at least to some degree over a corner, and the USE_BOX trace, starting from let's say the middle of the player position, will then hit the ground on the downward trace.
As 3run already said this is a tricky one to solve - or at least to find workarounds -, especially in three dimensions. Maybe a separate small collision hull (entity) near the feet with a separate c_move could do the trick (then limit the player's movement/ c_move call accordingly).