Basically, what their script does is load an iframe wherever you put the code. If you want it on every page, it should be placed in your theme's template. Their shoutboxes are about 220px wide. So... create a div exactly that wide and position it on the page with css or inline styling. Be sure that wherever you position your div at it won't cover up the ability to log in to your site!
So something like:
Code:
<div style='width: 220px; position: absolute; bottom: 0; left: 0px'>
<script type="text/javascript" src="//www4.smartchatbox.com/shoutbox/start.php?key=705821669"></script>
</div>
If you understand absolute/fixed positioning with css it'll help a lot.