Theme Development for a Newbie

Looking at your code (as i do not see any live link to look at). I will assume you are outputting anchor tag inside a button tag (making the HTML invalid). Try the function below instead. Notice that I changed the html from button to anchor tag.

    /**
     * Add Login/Logout to Menu
     */
    function add_login_logout_link($items, $args) {

        $redirect_to = home_url();

        if( $args->theme_location == 'primary' ) {
            $loginoutlink = wp_loginout('index.php', false);

            $link = wp_logout_url($redirect_to);
            $label="Logout";

            if(!is_user_logged_in()){ // user not logged in, show login url
                $link = wp_login_url( $redirect_to );
                $label="Login";
            }

            $items .= '<li><a class="btn btn-default" href="'. $link .'">'. $label .'</a></li>';

        }

        return $items;
    }

add_filter('wp_nav_menu_items', 'add_login_logout_link', 10, 2);

Using Output Buffer

Only use output buffer when you have to. This has potential to create unusal behavior. Here is an example.