やり方

      this.$auth.loginWithRedirect({
        hoge: 'nyaaa'
      });

と書けば Ruleのcontext.request.queryには

{
  "protocol": "oauth2",
  "client_id": "xxx",
  "redirect_uri": "http://localhost:3000",
  "hoge": "nyaaa",
  "scope": "openid profile email",
  "response_type": "code",
  "response_mode": "query",
  "state": "TnM2UTRhR2ZvanBna28zSklZSjdxNFlfRGpoMDMtY3FKdX5TVUl6ZXduQg==",
  "nonce": "vhGKxvRCUezZ6eLlpO-.fD17O0xXZd5JJoFr~37~S9Q",
  "code_challenge": "",
  "code_challenge_method": "S256",
  "auth0Client": "eyJuYW1lIjoiYXV0aDAtc3BhLWpzIiwidmVyc2lvbiI6IjEuMi4zIn0=",
  "connection": "Username-Password-Authentication",
  "realm": "Username-Password-Authentication"
}

hoge:"nyaaa"のKey Valueが追加される！