The answer is to register the desired binaries with the binfmt-support service.
Studying the difference between what is on my 16.04 installations and my 14.04 installation, I found that the previous Ubuntu version had a larger list when running the command update-binfmts --display.
The list includes a jarwrapper.
Installing the jarwrapper resolved the issue:
$ sudo apt install jarwrapper
The binfmt-support is a service that c an be configured to detect what is needed for running various type of files, including calling Wine to run Windows exe files.
Details for using the binfmt-support service can be found by studying the update-binfmts man pages.
Curious as to why the default support was dropped. But I'm glad to be able to manually restore this support to my computers.
Note
By the way, the program type isn't determined by the extension. It's determined by the application binary header. So the application can be named anything, without an extension.