I have a feeling this is because of an update to Linux, but I will go to bed after ensuring the program is running, wake up in the morning and it will be gone!

To try and resolve for now, I have performed an uninstall/purge/install of BOINC and it worked fine on the first boot-up.

The system does not run with an active screensaver and the sleep/hibernate flags are off.

I'm not experienced with Linux - the only reason I built this little 4-core is so it would run BOINC next to my 6-core.

Any help would be greatly appreciated!

Check the logs for errors related to: boinc, xorg
As Mint is a Debian derivative, the logs (logfiles) are most likely in folder /var/log/ unless you configured things to log to some other folder.
If it's the manager (the GUI, not BOINC client) that's disappearing, a reasonable first guess would be something related to (or reported by) the X subsystem. X logs to Xorg.0.log on my Ubuntu system, probably on Mint too. BTW, X is the video sub-system. It ultimately provides the desktop and GUI's, for example, BOINC Manager.
You should also check /var/log/ for BOINC's own logs and search them for error messages.
Finally, the error messages you're looking for might even percolate up to syslog or other logfiles.

My Mint 18.3 has: var/lib/boinc-client

BOINC is version 7.6.31 from the Ubuntu site.

And it doesn't disappear. :)

A thought:

Find where the files are.
If/when it disappears again, have another look to see if the files are still there, or they too have vanished.