There are two features that will force an event to require login, even if it's marked as public.
1) Overlapping event check. If you have this enabled in configuration, ALL events will require login.
2) Prerequisite events/categories. With an event, you can select events or categories as prerequisites, meaning the user has to have registered for that before registering for this event. Using that feature ill require login. If you accidentally selected prereq items not knowing what it was, you can de-select using CTRL-Click (Command-Click on a Mac).
I am attaching a screenshot of the event configuration. There are no pre-requisites and its not checking for overlapping events but I still get the same error:
There is no other way this would happen. The overlapping event check I'm referring to is in the main configuration area, not in the event setup. Check that too. If you can't find it, log a support ticket so I can check your site and see what setting is causing this.