Taxonomy Checkbox Admin Panel

Change the name of the checkboxes fromfirm to firm[].

Then, when you go to save the checkboxes

$terms = $_POST['firm'];

will give you an array of term slugs that can be checked/sanitized prior to using wp_set_object_terms to add the terms…

wp_set_object_terms( $user_id, $terms, 'firm', false);