Anyway, it all depends on how you want to implement it. Using Facebook on your website is ofcouse way easier to implement than for example in the source of GunZ. I don't know how that works but one way you can do it is: use the Facebook API to let users login on your website and if they don't have a GunZ account yet, create one for them (choose username and passwords etc). All you have to do is link their Facebook ID to their account.
+1 to this. Let them register with Facebook, or let them link a preexisting account to Facebook. Store facebook token alongside the account in your pserver's database, and whenever someone logs in again with Facebook, you can consider it the same as a normal login to your pserver.
I see the thought pattern behind thinking you may need Chromium for this. Unless I missed something, it looks like the only way for "Facebook login" to work is by literally directing the user to the appropriate facebook.com OAuth page. This is not uncommon for OAuth logins - Facebook wants to consume the user's facebook username/password; they definitely don't want to receive a user's Facebook user/pass from your app (they don't want your app to know about a Facebook user's user/pass).
You can allow Facebook login by opening the appropriate facebook.com OAuth page in the user's default browser from your game client or launcher. When they login on Facebook, you can have a web server set up on your end (for Facebook's redirect URI) that waits for Facebook to ping it, letting you know that a specific facebook user wants to log in to your server. Then your game client or launcher can automatically log them in to the appropriate account (i.e. the account on your pserver that the Facebook user is linked to).
The most unfortunate part of the user experience is requiring the user to login to Facebook from facebook.com. You could try to redirect them back to a URL that opens the launcher for them, or you could try to redirect them to a URL that just tells them to check their launcher (which should auto log them in after receiving the user's Facebook oauth info).
The hardest part may be automatically logging them in once they login via the facebook.com oauth link. You need something in your game client/launcher listening for a message from your web server (the redirect URI) that tells it to log in with X account, and you need to do so securely.