Require specific access for certain pages with only one input field
so the characters can be random or need to be verified against some database table entries?

best thing would be to have a custom code/solution for this, use ctools popup and custom javascript to check for the right cookies on the selected pages, we can add a field defining the setting whether the page requires verification or not, and overridden node tpl (template) for that particular content type which can contain drupal_add_js or inline javascript based on the field condition. Adding a module and customising will add more pain in my opinion.