Working fine for me. Could be specific to you since I don't see any recent server troubles... If you are on steam try verifying game integrity and/or logging in fresh instead of using the remember me function.
I'd advise contacting support or re-installing, there seems to be some trouble connecting you to the servers.
Edit: Armoredmeat also provides good suggestions with the firewall and that.
"ServiceUnavailable" probably means exactly that - the logon server was unavailable, ie down. Happens from time to time. Try again after 15 minutes or so, if it's still the same contact support.
In the launcher, go to options->validate game files. Crashes could result in some files not being in a proper state, which may result in all kinds of weirdness. Using this option makes the launcher check if everything is ok on your end.
Disable windows firewall and disable any firewall/anti virus software temporarily. It has to be a firewall issue. Bypass your router and go straight to your modem if you can too