Barn Door with no header
I suppose "no header" would mean that it's a passageway, clear from floor to ceiling, rather than having a foot and a half of wall overhead as a doorway would. Rather than saying the walls aren't "square," maybe it would be more clear to say there's a jog or offset between the walls on either side of the passage -- they aren't collinear.

Mounting a barn door here shouldn't be terrible. Barn doors hang from and slide along an overhead track. Choose one with a track that looks stiff enough it won't deflect sideways nor sag in the span across the passageway.

The offset between the two walls can be dealt with by shimming. Blocks or a strip of wood could be mounted to the "background" wall to create a surface about the same depth as the "foreground" wall; then the door hardware could be mounted to those blocks on the one side and directly to the wall on the other side. That's just one idea. It might be possible to use thicker blocks on one wall and thinner blocks on the other so as to keep some consistency on both sides. Or maybe the track mounting hardware for the specific door you'll choose would be adjustable so that the space between the door slab and the wall is adjustable. If so, simply adjust it far from the wall on one side and near to the wall on the other side.