Is there a way to configure a Stream to not appear in the Outlook or Chrome extensions for sharing, such as a hidden Stream that I don't want to be shareable?
Yes, you can prevent any Stream from appearing in the Outlook and Chrome extensions by configuring a Stream Option.
To do that, log in to Uberflip and navigate to the Stream you want to hide from the extension (Hubs > Content > click on the Stream). When you're viewing the Stream, click on the Options tab at the top, then scroll to the bottom of the page. Set the Show in Extensions setting to off (grey). Your change will be saved automatically, and the Stream will no longer appear in the extensions.