Update:
An out-of-band optional update is now available for the following versions of Windows 10 on the Microsoft Update Catalog to address a known issue in which certain printers may be unable to print after installing updates released June 9, 2020.
https://support.microsoft.com/en-in/help/4567512/windows-10-update-kb4567512
https://docs.microsoft.com/en-us/windows/release-information/windows-message-center#437
Success. My Brother MFC-240C USB Printer/Scanner is now working properly again.
Strange, though, under "Devices and Printers", my printer it is listed under "Unspecified", instead of "Printers".
Instructions for those who are not computer experts:
The link given will take you to the Windows message center. In the applicable message there, you will need to click on the link to the KB for the version of Windows 10 you have.
If you don't know which version of Windows 10 you have, then run "C:\Windows\System32\winver.exe" at the Win+R prompt.
Once you have opened the webpage for the correct KB, then scroll down to "Install this update". If you are running a regular computer (not a server), click on the link for the "Microsoft Update Catalog".
That will take you to a page that contain links for all versions of the KB, so you will need to find the version for you. For most people, it will be the last one:
"2020-06 Cumulative Update for Windows 10 Version 1903 for x64-based Systems (KB4567512)"
If you have a 64 bit computer. Click the Download button, and choose to save the file.
If you have uninstalled the troublesome update, KB4560960, then install it again. Reboot.
Then run the file you just downloaded. Reboot again. Then test your printer.