Hmmmm - Did the SSL certificate expire on the server? They do over time and some browsers are "particular" about who they will talk to.
If so - a fresh certificate would solve the problem.
IMO (I do this - sort of - for a living) since the problem happened to a wide cross-section of the users that the core issue should be on the central common point and not be an issue related to the user / browser end.
Presuming that there have been no software updates on the server and that the underlying hosting / connectivity provider hasn't changed - a safe assumption I think - that leaves a database corruption issue or a communications issue. Expiring certificates fall under the latter.
I've noticed (and ignored) that browsing the site under Chrome I get a variety of info / warnings about security.
Just my 2 cents.
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