Since the full Jordan curve theorem is quite subtle, it might be worth pointing out that theorem in question reduces to the Jordan curve theorem for polygons, which is easier.

Suppose on the contrary that the curves $A,B$ joining opposite corners do not meet. Since $A,B$ are closed sets, their minimum distance apart is some $\varepsilon>0$. By compactness, each of $A,B$ can be partitioned into finitely many arcs, each of which lies in a disk of diameter $<\varepsilon/3$. Then, by a homotopy inside each disk we can replace $A,B$ by polygons that join the opposite corners of the square and are still disjoint.

Also, we can replace $A',B'$ by simple polygons $A'',B''$ by omitting loops. Now we can close $A''$ to a polygon, and $B''$ goes from its "inside" to "outside" without meeting it, contrary to the Jordan curve theorem for polygons.

Since the full Jordan curve theorem is quite subtle, it might be worth pointing out that theorem in question reduces to the Jordan curve theorem for polygons, which is easier.

Suppose on the contrary that the curves $A,B$ joining opposite corners do not meet. Since $A,B$ are closed sets, their minimum distance apart is some $\varepsilon>0$. By compactness, each of $A,B$ can be partitioned into finitely many arcs, each of which lies in a disk of diameter $<\varepsilon/3$. Then, by a homotopy inside each disk we can replace $A,B$ by polygonal paths $A',B'$ that join the opposite corners of the square and are still disjoint.

Also, we can replace $A',B'$ by simple polygonal paths $A'',B''$ by omitting loops. Now we can close $A''$ to a polygon, and $B''$ goes from its "inside" to "outside" without meeting it, contrary to the Jordan curve theorem for polygons.