Stephanus,
We'd never recommend 777 even on a Linux system. I'd really recommend fixing your permission issues, as Facebook needs to be able to write and read from the filesystem during normal operation (downloading avatars, etc). If you can't install it, I think you'll run into other issues once you manually install it. Have you tried the enabling the FTP system within Joomla and using that to install? It may help.
If you need to install manually, you can try to unpack the component to it's administrator and component directories. Then, in the /administrator/components/com_jfbconnect/packages directory, you'll find 4 more files (3 plugins and a module) that will need to be unzipped and moved to their proper directories. After you do a discover, all 5 things should install properly. Again though, I think you may run into some other problems after that. Hopefully not.
Should you need anything else, just let us know,
Alex