Did you try option of separate email address? Also on same you can restrict accept email from or access to profiles?
Other option do not share the email address with those?
The most likely fix is to restrict senders from the Email side and say you have to be a member of a Group. Not sure how advanced your directory tools are but if you have dynamic groups, they may be able to have the group automatically add users that are in "Support" department(s).
If you need to do something in Remedy, you can check salesforce setup - Email Services. You can change who to accept email from. Blank means all. But you can list domains, or specific email addresses (very difficult to maintain unless you have a small support group).
Is the issue resolved?
Sadly no - I haven't been able to find a suitable resolution to this. The answer posted by Joshua would work if you had a small amount of external customers but would be too unmanageable with the amount of email addresses that would need to be added. I need someway of adding the function to the user account on SalesForce. Should be a simple checkbox that either "Allows" or "Restricts" users to create tickets with remedyforce.
Please create a Support case for this query.