public void setStatusLabel(String statusEntered) {
    //Shows the variable statusEntered has been received
    System.out.println(statusEntered);

    //Not working
    status_label.setText(statusEntered);

    //Used this to check if the label receives the data. It does.
    String status = status_label.getText();
    System.out.println(status);
}

//GUI Class reference
MainWindow mainwindow = new MainWindow();

public void connect(){
    Connection conn = null;
    try {
        String userName = "root";
        String password = "";
        String url = "jdbc:mysql://localhost:3306";
        Class.forName("com.mysql.jdbc.Driver").newInstance();
        conn = DriverManager.getConnection(url, userName, password);

        //This works
        System.out.println("Connection Established");
        //The issue is with this guy
        mainwindow.setStatusLabel("Connection");
    }
    catch(Exception e) {
        System.err.println("Failed to connect to database");
        mainwindow.setStatusLabel("No connection");
    }
}

public static void main(String[] args) {
    java.awt.EventQueue.invokeLater(new Runnable() {
        public void run() {
            new MainWindow().setVisible(true);
        }
    });
}

public class DatabaseConnection {
   // MainWindow mainwindow = new MainWindow();  *** don't do this ***
   MainWindow mainwindow;

   public DatabaseConnection(MainWindow mainwindow) {
      this.mainwindow = mainwindow; // pass in the reference in the constructor
   }

   public void connect() {
      Connection conn = null;

      // ... etc

      // now we can call methods on the actual visualized object
      mainwindow.setStatusLabel("Connection");
   }

public class MainWindows extends JFrame {

    private JLabel status_label;
    public MainWindows(){
        status_label  = new JLabel("Status");
    }

    public void setStatusLabel(String staus){
        status_label.setText(staus);
    }
    /*.
     *
     * your code for visualizing
    .
    .
    .
    .
    .
    .*/

    public static void main(String args[]){
        MainWindows mw = new MainWindows();

        //this will return true if the connection will estblished and will false if not
        mw.setStatusLabel("Connection established "+ new DatabaseConnection().connect());

    }

}

public class DatabaseConnection{


        public boolean connect(){
            Connection conn = null;


            // adding this new variable to get connection status;
            boolean returnStatus = false;  //default return false if connection established it will true .
            try {
                String userName = "root";
                String password = "";
                String url = "jdbc:mysql://localhost:3306";
                Class.forName("com.mysql.jdbc.Driver").newInstance();
                conn = DriverManager.getConnection(url, userName, password);

                //This works
                System.out.println("Connection Established");
                //The issue is with this guy
                returnStatus = true;  // making status true if connection is established

                return returnStatus;
            }
            catch(Exception e) {
                System.err.println("Failed to connect to database");
                return returnStatus;
            }
        }

    }