It is actually quite simple. You need to go to the location, where you have PHP-fusion installed, go to infusions -> shoutbox_panel and open shoutbox_panel.php with a text-editor.

Now, just beneath the copyright-declaration, and one single if-line, you find the line
openside($locale['120']);
Make a space underneath it, and in that space type this code:
$settings['guestposts'] = "1";
Now, everyone should be allowed to post in your shoutbox.

To only allow members again, go to the file and delete the single line of code you entered.