Example form is linked to a database table, but won't be saved in this example.
When you submit the form, the value is presented with debug_dump(), so strings are quoted and array values listed.
<?php
$field = new form_field_checkbox($form, 'Check');
$field->db_field_set('check');
$field->text_values_set('true', 'false'); // Remove to return a boolean.
?>