A comment by the dissident frogman on Houseworks
If it's not the browser, it could be some kind of "web cleaning" proxy (like the Proxomitron, which I use myself - you just need to add trusted sites [like the Me! I am trusted, am I not?] to its "Bypass" list) or some Ad removing software that's a bit too genocidal on the script cleaning side.
I'm just guessing here, as I could not reproduce the bug so far. Being such a sensitive feature, I've tested this on a wide array of browsers and operating systems to make (as) sure (as possible) it will work. Which doesn't mean I've got everything covered, of course.
We'll work it out. I'll add a straightforward, 'old school' backup link to the alternate login page — but in the meantime, if you could let me know which browser you're using, that might shed a light on the origin of the problem.