Edit section and deselect Ticket type
And unpublish (or delete) Private Support category
In fact you have to know that this is not recommended. There is a war for users in internet. The one of the key to success is not abuse your users. And asking for registration just to post message or to say something in comments I consider to be an abuse. You have to have very strong reasons to force everyone to register.
If I have not convinced you or you have that strong reason, open Topic type settings and change Who can submit to Registered
Find and follow Moderators link in your menu
And click New button.