How can I escape the brackets so they are seen as literal characters
The problem wasn't the brackets—I was able to set the rule by referencing the nested #name and #value keys:

_clientside_validation_set_required(
      $element['field_first_name']['und']['0']['value']['#name'],
      $element['field_first_name']['und']['0']['value']['#title'],
      isset($element['field_first_name']['und']['0']['value']['#required']) ? $element['field_first_name']['und']['0']['value']['#required'] : FALSE,
      $js_rules,
      t('Please enter your @field_name.', array('@field_name' =&gt; strtolower($element['field_first_name']['und']['0']['value']['#title'])))
    );