This option is not available from the Properties screen for security reasons. The "Enable query for CRM REST API" option provides access to the query's results when you use the CMS REST API for your website. When you edit a query with this option selected, any change to it affects your website. For example, the webiste reflects the security rights of the last user to save the query. If that user has rights to view A and C in query results but not B, website users can only view A and C. IF that user has rights to view A, B, and C in query results, website users can also view all three.
To work around this, you can re-edit the query and make the change to the option on the Save Options tab.