>I have added a user in my database permissions so this will be the one to use in the connection string from the application. However, I need to do a setting somewhere to define my database to not accept integrated security. Presently, even if I added a user in the database permissions, someone can still use a connection string with integrated security to connect to our database. How do I disable that?

Normally you have to distinctly set up users (even integrated users) to have access to your databases. I am unsure as you why it just works by default for you.