The boolean modifier will not work properly for non manifold objects (not solid), nor will it join properly 2 objects that do not overlap in some way.
Even if you solidify your “open” - non manifold boxes and overlap them so that the boolean modifier works you would have to manually clean up the topology after applying it.
Your best bet is to join the meshes (Ctrl J) and then in edit mode manually add edge loops slide them into place with the vertex snap option and then merge double verts.
After joining the objects add these edge loops:
Then merge these vertices:
Note that you will still end up with a non manifold mesh that will be no good for smooth shading.
Edit: If you want to make the boxes manifold you should model them something like this. (I also joined them together with the manual method that I described earlier, in this case you also have to delete the faces that are “inside” the joint.)
If you add a bevel modifier to it you will be able to smooth shade this one.