Healthcare users will only be able to see the free/busy information of the Office 365 users. Publishing: this options allows you to publish a calendar for use on a web page, for a non-Office365 user to see, etc. Sharing: this option allows you to specify who can view your calendar, the details they can see and if they have editing or delegate rights. Giving Permissions: this option allows you to specify who can view your calendar, the details they can see and if they have editing rights.
NOTE: if you are trying to share a resource, see Office 365 Resource Management
There are several options for sharing your personal calendar with other users - sharing, publishing, giving permissions.
