Hi JWarren,
Unfortunately there is no way to create a Global Case within Method, a case can only be related to one customer/contact. You could however try adding a specific "Tag" to each of these customers when they contact you regarding the system being down. From here you can go to the Campaigns Tab, then ListBuilder Tab Link. Using ListBuilder will allow you to make a list of all your users and filter through the tags you have set. This allows you to create a list that you can work off, you can also send out a mass email from here targeting these Customers that called in during the down time. I have provided a link here, which explains how to setup Listbuilder if this does turn out to be an option for you.
For information on working with Tags in Method, you can check this Help Center Article.
-Ben