K,
There's definitely something strange happening on your site. Sometimes, IE will continuously say it's trying to load the page. Either the status bar will say "Waiting on
www.sintagewpareas.gr/dev/... or the green bar in the middle of the status bar will just keep moving. The Facebook button won't appear until the page completely loads, so this is something to look into, but I don't think it's the main problem.
Unfortunately, I don't know what the main issue is. Oddly, after the page looks to have fully loaded, if I click to another page, the Login With Facebook button will show briefly. I think there's a Javascript conflict on your page, but I can't figure out where it's coming from. Can you try disabling some of your Javascript based modules (the menu on the left, the slideshow, etc), and see if disabling any of them makes the Login button? If you can narrow it down, that will help us figure out a solution.
Thanks,
Alex