Mike,
I just tried to login using IE8. The Login button properly worked and brought me to the normal JFBConnect Login/Register page. On the home page, before I clicked Login, I was getting the following Javascript error in IE. I don't always see it on the home page (or other pages) which may explain if the problem is intermittent. Facebook is very reliant on Javascript, and if there are errors on the page, it could be causing the issues you're seeing. The error was located in the bottom left with a little yellow
Webpage error details
User Agent: Mozilla/4.0 (compatible; MSIE 8.0; Windows NT 6.1; WOW64; Trident/4.0; SLCC2; .NET CLR 2.0.50727; .NET CLR 3.5.30729; .NET CLR 3.0.30729; Media Center PC 6.0; .NET4.0C; InfoPath.2; IPH 1.1.21.4019)
Timestamp: Fri, 17 Jun 2011 01:35:26 UTC
Message: 'this.el' is null or not an object
Line: 105
Char: 26
Code: 0
URI: www.medlink-uk.net/
Message: 'this.el' is null or not an object
Line: 106
Char: 26
Code: 0
URI: www.medlink-uk.net/
I'm seeing all the same behavior that you are though, after the first Login click, things start to work better with the menus. However, if I close my browser and start again, it's not just the first Login click that fixes things. On this 2nd attempt, I clicked the "Schools" link, and then went back to the homepage and things were fine again. So it seems like simply moving through the site fixes things.
I'm not seeing the Javascript error above in other browsers, and when I view the source, that line doesn't have anything like "this.el" on it. Since, when you view the HTML it re-fetches it from the server, it's likely that whatever's causing the error is not there anymore because (like above) we've gone to a new page.
I'm not sure what it is that's causing it, but do you have any tracking software, or things that would try to set a cookie or session variable for the user on their first visit? It seems like that's where to look, but it's not a Facebook thing from what we can tell.. the Javascript error that's causing the problem is simply causing Facebook to have problems.
Digging through the HTML on your site, I notice a few things that could be problematic:
1) Joomla loads Mootools, a Javascript library. It includes version 1.12. Your template also appears to be loading the older version 1.1 of it through the following line: /templates/at_tenota/at_menu/at_menu/mootools.v1.1.js
2) The at_menu in general appears to be setting cookies and doing other things that seem odd for a menu to be doing. This could be absolutely normal and fine, but also goes along with the above where I mention setting cookies or other 1-time session variables. The file that's doing this (though I don't know what/if anything is calling it):
templates/at_tenota/at_menu/at_menu/at.menu.js
You may want to try a different template (temporarily) to see if that fixes the issue.. which will help narrow things down. Then, from there, start disabling different modules, plugins, or template features to see if there's a difference. Also, you may want to contact your template provider to see if they have a version that doesn't include an older version of Mootools or, in general, has other bugfixes.
Really don't know if I'm barking up the wrong tree with these suggestions, but hope they help you narrow down the possible conflict.
Thanks,
Alex