FFXIV Launcher Error

[ThatOnePerson]

Whenever I try to log in on the launcher, I get a "FFXIV Launcher Error". In the dialog box, it says:

Unable to complete version check.
[30410][30607][10009][10020]

If I try logging in multiple times before restarting the launcher, the list of numbered codes repeats itself once for each try, which is sort of fun.

I know there have been previous topics on this (usually with a [10019] instead of my [10020]), but I've tried everything on those topics short of reinstalling the game. It's only after deleting boot files, restarting my router, updating my browsers, and so forth that I can even get the launcher to a place where I can type in my username and password. Any other ideas, before I commit myself to uninstalling and reinstalling?

EDIT: After restarting my computer, I was able to log on, but then got a 2002 error while trying to access the Aether data center. While trying to fix that, I restarted my router. After restarting the router, I get a different "FFXIV Boot Error":

Version check failed. Please confirm your network settings. If this does not solve the problem, you may need to whitelist FINAL FANTASY XIV in your security software.
For further assistance, please visit the SQUARE ENIX Support Center.
http://sqex.to/x4w
[30605][10009][10020]

On the bright side, I seem to have dropped the [30410] at this point. I'm assuming that that's a positive.