TwitterWebService.gs

'use strict';

function getInstance(consumer_key, consumer_secret) {
  return new TwitterWebService_(consumer_key, consumer_secret);
}

var TwitterWebService_ = function (consumer_key, consumer_secret) {
  this.consumer_key    = consumer_key;
  this.consumer_secret = consumer_secret;
}

TwitterWebService_.prototype.getService = function() {
  return OAuth1.createService('Twitter')
    .setAccessTokenUrl('https://api.twitter.com/oauth/access_token')
    .setRequestTokenUrl('https://api.twitter.com/oauth/request_token')
    .setAuthorizationUrl('https://api.twitter.com/oauth/authorize')
    .setConsumerKey(this.consumer_key)
    .setConsumerSecret(this.consumer_secret)
    .setCallbackFunction('authCallback')
    .setPropertyStore(PropertiesService.getUserProperties())
}

TwitterWebService_.prototype.authorize = function() {
  var service = this.getService();
  if (service.hasAccess()) {
    Logger.log('Already authorized');
  } else {
    var authorizationUrl = service.authorize();
    Logger.log('Open the following URL and re-run the script: %s', authorizationUrl);
  }
}

TwitterWebService_.prototype.reset = function() {
  var service = this.getService();
  service.reset();
}

TwitterWebService_.prototype.authCallback = function(request) {
  var service      = this.getService();
  var isAuthorized = service.handleCallback(request);
  var mimeType     = ContentService.MimeType.TEXT;
  if (isAuthorized) {
    return ContentService.createTextOutput('Success').setMimeType(mimeType);
  } else {
    return ContentService.createTextOutput('Denied').setMimeType(mimeType);
  }
}

Forked to update OAuth1 library v14 due to Project key EoS. You can use following Script ID to include the library.
1rgo8rXsxi1DxI_5Xgo_t3irTw1Y5cxl2mGSkbozKsSXf2E_KBBPC3xTF
The sample code written by YoshiyukiHirano is here.

2018年6月からTwitter社がAPIの仕様を変更して、Callback URLの設定を必須にしたのですが、このスクリプトのどこを改造したら良いでしょうか。

恐らくこのスクリプト自体に修正は不要だと思われます。
Callback URLは、"https://script.google.com/macros/d/[GASのスクリプトID]/usercallback” にて設定した所正しく動いています。