Not that it helps you any, but I'm pretty certain WN either uses a "Man-in-the-Middle" certificate when you're using the in-flight WiFi, or has different certificates for southwest.com aboard (I guess determined by origin IP address) than when on the ground; I'm not an expert on the certificate chain, but there's times I've had to switch browsers while trying to log into the WiFi due to certificate errors on the login page (and I refuse to ignore that error and log in anyway, it could be someone spoofing southwest.com to collect logins).

What this means is maybe the app is trying to validate against the "wrong" certificates (or can't verify the ones it's trying to use)- but that's just my SWAG.