Please tell me what you have set up in your Facebook application for App Domains (in Facebook dev center > app > Settings > Basic area) and Valid OAuth Redirect URIs (Facebook dev center > App > Products > Facebook Login > Settings).
The redirect URIs should follow the format listed in
our application setup guide
. In your case with a subdomain, it would be like:
https://subdomain.yourdomain.com/index.php?option=com_jfbconnect&task=authenticate.callback&provider=facebook
https://www.subdomain.yourdomain.com/index.php?option=com_jfbconnect&task=authenticate.callback&provider=facebook
All variations with https/http and www/non-www should be listed in redirect URIs.
If that doesn't help, please tell me what you have set up for your whitelisted IPs/etc. It might be worthwhile to leave all the whitelist settings blank until you have the app set up properly.
My main testing site is on a subdomain, so JFBConnect definitely works. Let me know and we'll go from there.
-Melissa