Yesterday, ccleanercloud stopped working...

tr...And my computers are still showing as offline! I've tried restarting from the tray-icon, but their status is only 'connecting...'

As CCleaner Cloud is a business edition then you have priority support and should contact them using this form:

https://support.ccleanercloud.com/hc/en-us/requests/new

That's from the help pages that are specific to CCleaner Cloud, you might want to bookmark them: https://support.ccleanercloud.com/hc/en-us

same with me. a number of my monitored machines are showing offline and displaying "connecting" in the tray icon.

As it's happening to more than one user then it sounds as if there is a server issue.

Again, the CCleaner Cloud support team would be your contact point to report the problem.

Automagically it started working again!!!

I guess that it was a server problem and the faulty server has been fixed then?