Adam,
That should have definitely worked based on how Joomla functions. Do you have something on your site that tries to 'force' all files to load over SSL either in the .htaccess or plugin for Joomla? If so, that may be causing some issues.
In general, the JURI::base function asks Joomla what the 'root' URL for your site is, and it should include the https. It's not on yours. You can, if you want, simply remove the JURI::base() line and just make it look like:
$newText = '<link rel="stylesheet" href="/components/com_jfbconnect/assets/jfbconnect.css" type="text/css" />' . $newText;
Unfortunately though, you'll need to make that change with each new release as that shouldn't be necessary (and isn't the right way to do it).
Keep us posted, and good luck,
Alex