It may not be the desired way but you can reset all user profiles to switch over from 'Everyone' to 'Registered' or you can set your site to lockdown mode.

An option to set default site wide profile setting is preferred though.

I believed if you are in Lockdown mode, Everyone is not an option in most of the Share privacy, I'm just not sure which areas I will have to take note of it. However, Default Privacy settings in Profiles are just default privacy templates and Users will be able to override it in their Privacy Preferences - (Correct me if I wrong on this the last time I check it does that). My wish is the Privacy choices (Everyone, Registered Users, Friends, Friends of Friends, etc.) be made more granular for Site Admin to control. Just like the Joomla ACL where Site Admin can also define their own. Anyways, just an idea

Hello,

Unfortunately there's no easy way to remove this but you can hack the theme file if you need to customize this. The theme file is located in /components/com_easysocial/themes/privacy/default.php

Hello Mark, is possible to remove this option in the future for private sites?

Hello Jasonargo,

To disallow your user from setting 'everyone' in the privacy, the only way now is to enable the lockdown mode in your EasySocial. Once you've the lockdown mode enabled, then head to your EasySocial backend under privacy and set all your privacy rule the default option to 'member'

Let me know if the above method work for you or not.
PS: Once you've enabled lockdown mode, remember to enable the 'allow registration in lockdown mode' as well

Hope this help and have a nice day