Is it possible to alter the cacheability of a specific entity either programatically or via configuration
I just came up with a solution, but I'm not sure it's the "right" way to do it. 

Anything entity that implements RefinableCacheableDependencyTrait will allow you to introduce additional dependencies with a call to addCacheableDependency(). I sort of hoped there might be a predefined dependency that was hardcoded to no cache. I didn't find such a thing. What I did find is that if the argument to that method is not an object that implements CacheableDependencyInterface, then it makes it uncacheable. So, here's what I ended up with:

  public function build(RouteMatchInterface $route_match) {
    $breadcrumb = new \Drupal\Core\Breadcrumb\Breadcrumb();
    $breadcrumb-&gt;addLink(Link::createFromRoute($this-&gt;t('TESTING'), '&lt;front&gt;'));
    $breadcrumb-&gt;addCacheableDependency(null);
    return $breadcrumb;
  }