Assuming the boxes are in cells. If not you can make new boxes but the must be in cell coordinates. Use GDALinfo to get the coordinates:
Upper Left ( 0.0, 0.0)
Lower Left ( 0.0, 220.0)
Upper Right ( 220.0, 0.0)
Lower Right ( 220.0, 220.0)
Center ( 110.0, 110.0)
This is part of the GDALinfo on an ungeoreferenced image (220 pix by 220 pix). Have a look at How do I create point features with exact coordinates? to create a box with those exact coordinates.
Georeference the images using the georeferencer and then recycle the control points to transform the box. see How to georeference a vector layer with control points?.
You may need to do some work in notepad or similar to convert the control points from the image georeference format to vector georeference format.
After the images and boxes have been georeferenced you can extract the common area between the shapefiles using OGR2OGR with
-clipsrclayer option or if you don't like command line look at this How to crop Shapefiles in QGIS? to crop one bounding box with anothers' bounding box.
Extract the common areas from each georeferenced image as seen in this post Clipping raster with vector boundaries using QGIS?
That should get you within a pixel or two of the true overlapping area. The result images will be georeferenced; if you translate to a worlded format (.bmp, .jpg) you can then delete the world file and they will be no longer georeferenced.
Disclaimer: I have only done this in ArcMap and I am assuming that QGIS will behave in the same manner based on the same inputs. It worked in ArcMap.