How to add HTML to a template only when user is logged out/ not registered

First of all, your shortcode usage code is incorrect, correct code is:

echo do_shortcode('[loginout_button]');

Now, if you want to echo the login/logout button only when user is not logged in, you can use the function is_user_logged_in(). There are two ways to do this

1). Conditional logic inside your shortcode.

add_shortcode('loginout_button','add_loginout_button');
function add_loginout_button() {
    if ( is_user_logged_in() ) {
        return '<div style="text-align:center;background-color:#7114B7;padding:15px;border-radius:50px;margin:20px;"><a href="">LOGIN/REGISTER</a></div>';
    }
    return false;
}

This is the recommended way because you can use the shortcode anywhere and get the expected result.

2) Wrap your shortcode usage code in the conditional tag

if ( is_user_logged_in() ) {
    echo do_shortcode('[loginout_button]');
}