same here. i wanted to play bf4 again after a while. first i couldnt because the "play" button in origin was greyed out. then my browsers (chrome, ff) kept telling me i needed to update the plugin.
i reinstalled origin, repaired and reinstalled bf4 yadda yadda, only to still get the same error message as OP.
i too tried moving the bf4 folder, redownloading it, pausing the download, moving the folder back to its old location.
still, not working. and just to clarify, i have not changed anything in my firewall or router. i can join bf3 servers just fine. the game wont even load when i turn my firewall off.
so, i am really looking forward to some help, since everything i found i have already tried. and again, it was working perfectly fine a few weeks ago.