This is what worked for ME (did the stars align and servers worked after ten times without doing this? Could be.) [I tried it in with my friend's computer as well, same thing]
Go to Program Files (x86) > SquareEnix > FINAL FANTASY XIV - A Realm Reborn
Right click on the boot folder and uncheck Read-only > Apply > Apply to folder, subfolder and files
Do the same thing to the game folder
Try to update again.

Hope it helps