Quick note: I finally managed to get the APEX Developer Addon signed by Mozilla and therefor working again in Firefox 43 upwards.
Please re-download and install, version of the Addon in Firefox is now 2.0.0.49. No changes for Chrome.
Apologies for the inconvenience and thanks for all those mails I got.
With Chrome Version 27 a security bug in the browser was fixed which caused the APEX Developer Addon to break.
Thanks to all sending me a reply about that, i’m currently working on a fix and hope to be able to provide that soon.
In the meantime please stay calm and use Firefox instead.
P.S.: no, the APEX Developer Addon didn’t exploit a security hole. It is just an unhappy coincidence that fixing one security problem broke our extension.