theme-my-login-custom.php

<?php
function tml_registration_errors( $errors ) {
    if ( empty( $_POST['country'] ) )
        $errors->add( 'empty_country', '<strong>ERROR</strong>: Please enter your country.' );
    return $errors;
}
add_filter( 'registration_errors', 'tml_registration_errors' );

function tml_user_register( $user_id ) {
    if ( !empty( $_POST['country'] ) )
        update_user_meta( $user_id, 'country', $_POST['country'] );
}
add_action( 'user_register', 'tml_user_register' );
?>