How do I add input a value to a Craft/Twig request? This might be a stretch.  

How do I (literally) modify the (Tableau API) javascript function below to print a specific fieldName ("Country") so that it can place the "Country" into the CraftCMS/Twig request below?

{% set results = craft.entries.section('posts').search('countryName:"Value Goes Here"') %}
You can't.

The Craft / Twig code runs server-side, before the page is delivered to the browser. The JavaScript runs client side, after the page is delivered to the browser.