Looking at the console, there are a lot of 404 errors on your site now.
Before clicking the button, there are a ton for images being loaded.
After clicking the Twitter button, here's the error:
Mixed Content: The page at 'www.XXXXX.com/en/?option=com_jfbconnect&...b2a882c1a41fd6379d=1' was loaded over HTTPS, but requested an insecure stylesheet 'fonts.googleapis.com/css?family=Open+Sans:400,300,600,700,800'. This request has been blocked; the content must be served over HTTPS.
Did you switch over to https recently, changed any settings related to that, etc?
-Melissa
Twitter login not working
