Excel closes without saving changes

Repair Office and verify the status.  to do so go to  Control Panel > Programs >  Programs and Features > Select the Office suite installed > Click on Change > Repair > Continue.  Try Quick repair first and if it didn't help try Complete repair.

Does the workbook contain a macro?

Couple things come to mind....any macros?  Probably not as you uninstalled and re-installed.

Check for virus/malware.

Do you see this if you open a new spreadsheet add some text then click on the red x?

Previous to reinstalling Office, I did the various repairs, and just tried again (just to be sure) -- No change.
There are no macros in the workbook -- As in the video, even if you open a new, blank workbook, the same problem occurs.  

This just issue started a week or so ago.  (So, before anyone asks -- I did not do the recent Windows 10 update to .1709)

--- thanks

Try starting Excel in safe mode and see if it continues,

start>run> excel /safe

How about creating a new User Account in Windows and retest.
Did you Try Windows System Restore ?

Make sure you have installed all security Updates.
https://support.office.com/en-us/article/Outlook-known-issues-in-the-June-2017-security-updates-3F6DBFFD-8505-492D-B19F-B3B89369ED9B

Ok ... interesting ...  
In  SAFE MODE, Excel does not have the problem.  It prompts me to save my changes, as it should.

Also, I noticed that in "regular" mode, when I clock the red "X" to close Excel, it takes a while for it to close -- the cursor spins for a few seconds, then excel closes. But, this does not happen in SAFE mode -- Once I click the "X", I get the prompt immediately, click "Don't Save", and it closes Excel immediately.

If none of the above fixed the issue then try to repair Excel from below link:
https://support.office.com/en-us/article/Repair-an-Office-application-7821d4b6-7c1d-4205-aa0e-a6b40c5bb88b

Thanks again!

Thanks for your feedback and I'm glad I could help.