private Firebase.Auth.FirebaseAuth auth;

:

// Use this for initialization
void Start () {
  auth = Firebase.Auth.FirebaseAuth.DefaultInstance;
}

 :

public void OnClickLoginButton()
{
  Text usernameField = GameObject.Find("UserNameText").GetComponent<Text>();
  Text passwordField = GameObject.Find("PasswordText").GetComponent<Text>();
  string username = usernameField.text;
  string password = passwordField.text;
	auth.CreateUserWithEmailAndPasswordAsync(username, password).ContinueWith(task0 => {
    if (!task0.IsCanceled && !task0.IsFaulted){
      Debug.Log(string.Format("Created {0}", usernameField.text));
      auth.SignInWithEmailAndPasswordAsync(username, password).ContinueWith(task1 =>{
        if(!task1.IsFaulted && !task1.IsCanceled){
					Debug.Log(string.Format("Logined {0}", usernameField.text));
          Firebase.Auth.FirebaseUser user = auth.CurrentUser;
          if(user != null){
							Debug.Log(string.Format("UserName is {0}", user.Email));
              Debug.Log(string.Format("UserID is {0}", user.UserId));
              // ここにログイン完了時の処理を入れる
					}
				}
			});
		}
    else{
      Debug.Log(string.Format("Creation failed {0}", usernameField.text));
    }
  });
}