Today, mysteriously, the desktop icons on my windows 10 machine have gray "x" and green checkboxes as an overlay. The state of the overlay seems to flip randomly from time to time.
WTF?
Part of your question can probably be answered here:
In other words, people who previously saw the green check boxes attributed them to either Microsoft OneDrive or locally installed Norton software.
As far as each gray "X" is concerned, there is a distinct possibility that they are symptomatic of the same root cause. From what I have seen, you can alleviate them by doing the following:
Right-click
on the desktop shortcut or file.Open with...
Always use this app to open .??? files
OK
button(Source)
One drive is adding it. It is simply because after an update one Drive decided to sync your desktop with other User folder like documents, pictures public etc.
So this is a backup overlay similar to what Dropbox used to add to the files in the folders being synced.
The only way to remove it is to to go onedrive setting or right click on desktop and click "choose folders to sync" and uncheck your desktop.
I kept it on I don't mind my desktop files getting synced, I have to just get used to the green check marks.
Refresh
to see if that eliminates the problem.