This is incredibly unexpected.
I wonder if it a javascript conflict?
May I suggest a few things to try:
I see you have prototype and scriptaculous libraries, the js for AjaxSearch, and the ieButtonFix.js file (which is being loaded twice for some reason???).
First, try getting rid of the javascript you are including manually in your template, then see if the login buttons work in FireFox.
Next, get rid of AjaxSearch and see if the buttons work in FireFox.
If none of that works, you could try using text links like this
<a href="[~[*id*]~]&service=login">Login</a>
<a href="[~[*id*]~]&service=forgot">Forgot your password?</a>
<a href="[~[*id*]~]&service=register">Sign up!</a>
Please post back your results.
-sD-
Dr. Scotty Delicious, DFPA.