Is there a way to overlay the borders of modern countries over the various maps of biblical events? It would immensely help for teaching to be able to turn on and off this feature.
Unfortunately no. We've chosen (so far) not to create editable maps.
You can use the context menu to export to PowerPoint or other programs, and then add text, lines, overlays, etc. there. But i recognize that's not the feature you're looking for.
Have you clicked on a location and opened in Google maps which can show modern boundaries?
That is what I do. But it is not as useful, because it does not retain the boundaries or ancient place names. It would be nice if the modern country boundaries and names were a layer that could be turned off and on. Modern major city names would be icing on the cake!