Earlier quoted context omitted.
POSTed values won't show up in typical server logs, whereas with GET their plaintext password would normally be logged on every request. And POST over HTTPS keeps the POSTed values hidden from those who might listen, whereas with GET over HTTP it would just be out there in the URL.
You have no way of knowing what they log and don't log. If their server logs are compromised, you should be assuming their username/password database was as well. And HTTPS requests are encrypted. The whole request, including the "GET /someurl&password=s33krit HTTP/1.1" part. As I said, using POST doesn't add any additional security to this.
That's why professional security audits will ding you for putting anything sensitive in a URL.