My boss started having this problem this morning. It may possibly be due to an update. The calendars appear to still be working fine (not 100% sure they're getting new updates) and I've tried deleting them and re-adding them and the problem persists. Outlook will pop up the Security prompt asking for login credentials, with the username of the person whose calendar is shared pre-filled in the user name section. If you click cancel everything continues to work but it prompts you again in a few minutes. 

These calendars have previously worked perfectly with no issues. 

Hi,

Please confirm if there is any changes in Shared calendar account and your boss mailbox.

Since the issue occurs after the update in your Boss's computer, please reboot the local machine. And recreate the Outlook profile to have a try. About opening shared calendar in new profile, please open the shared calendar by the following steps:

1. Switch to Calendar pane.

2. Click Open Calendar > Open Shared Calendar.

3. Click Name to select the user who shared the calendar to this user.

4. Then click OK to open the Calendar.

Check whether there is any error when open it. If there is any permission error, please check the folder permission in shared user side.

Regards,