I answered this question in Office 365 community forums so i though I might document it here as well.
You can Share a part of your site with external users .. they will see the links in quick launch and in global Navigation but once they click they will get access denied.
So this is how it is done -
1. You send an Invitation to a person
2. Once they accept it, delete the user from Site collection. You can clear them from all groups or you can just click on the user name in one of the groups or in libraries and then click “Delete User from Site collection”
3. Now when the external user logs in they will get Access Denied.
4. Next Go to one of your libraries the pages/library/list/calendar/tasklist and add the user with whatever rights you want to give them in that library…
Here is how you do that -
1. Go to one of the libraries and then click on Library tab and then Library Permissions.
2. Next in Permissions window click on Stop Inheriting Permissions.
3. Now you can customize who can get the permission for this library. You can now add the external user here with whatever permissions..
Please note once you add an external person in one of the libraries it gets added as Limited Access to rest of the list\libraries. This means they can click on the libraries (wont get access denied) but will have limited content visibility.. you will have to remove them with same “Stop inheriting permissions” way if you want to restrict external users to even get to one of the libraries.
External users this way would not be able to go to even site home page as they are restricted to where they got the access..