Trying to make Case Owner un-editable but no matter what I do it still shows up as having the edit button? This page layout pictured in the screenshots is the correct one.
You cannot make
Case Ownerfield read only, from your field level security.
Just navigate to 'Case Owner' field and click on
Set Field Level Security button. You will see read only checkboxes are disabled.
Though you try to assign Read-only from Page Layout, it will still show up.
If you have Edit permission on that object you will still edit Case Owner field.
Write a validation rule and restrict if user of specific profile try to change the Case owner.
Create a formula field which will show Case Owner Hyperlink type.
Create a separate page layout for those profiles and remove original Case Owner field from this layout and put this custom case owner field on this layout.