I am having the same issue, but in my case it seems to be related to SEF being enabled. I have verified this to be the case as disabling SEF under Plugins....Search - SEF (Note: Joomla's core plugin) resolved the issue.
NOTES:
With SEF enabled initially, the home page worked and rendered correctly, but any link(s) you clicked from there, did not. The solution from Joomla's forum was to edit the CONFIGURATION.PHP line that reads:
var $live_site = '';
to instead have the full URL like so:
var $live_site = '
www.yoursite.tld';
This also means, like in my case, a domain forwarder at the server level really forwards
www.yoursite.tld/somedirectory/ to really be the site root. This means that when a user clicks a link to the TLD, or a bookmark/favorite, or just types in
www.somesite.com, the server says "that really goes here to
www.somesite.com/somedirectory/" and just works automatically.
So, on another site that I was also the creator of, I do not use any subdirectory(ies) for the Joomla root, and Jsecure works perfectly. It also does not have SEF set up. So, at this point all I can say for CERTAINTY is that in my case, while a domain forward is being utilized, disabling the SEF plugin

corrects the issue. So at this point, can you say with certainty that Jsecure DOES work with SEF enabled? With or without another SEF product enabled such as SH404SEF? I have that too, and it is likewise disabled.....
If not, then you and/or I need to test this out by enabling SEF, edit the CONFIGURATION.PHP with the var $live_site='
www.somesite.com/somedirectory' and see if it works. If it does AND removing "somedirectory" from the URL in your browser still shows what is above (like list the directory, etc.) then try this out with a domain forward at the server level.....
Please advise.