The usual account is that the shape arose from packing cigars tightly enough to survive transport, the box itself squaring them off over weeks at sea. Whether that origin is historically accurate is argued; what is not in doubt is that the shape is now produced on purpose.
Does it change the smoke?
The claims made for it — a denser bunch, a slower burn, a different draw — are argued rather than demonstrated, and this encyclopedia does not adjudicate them. What is not contested is that it changes how the cigar sits in the hand and against the lip, and that a square cigar does not roll off a table.
Degrees of press
A soft or trunk press leaves rounded edges and a gently squared body; a firm box press gives distinct corners and flat faces. Both are the same operation done to different degrees.