I think jailshell has a problem with shell substitution. I worked around this by creating the command in php.
echo "<?php date_default_timezone_set('America/New_York'); passthru( '/usr/bin/mysqldump -uXXX -pYYY db_name > /home/mysql_backups/backup-'.date(DATE_ATOM));" | php -q
This worked for me inside a cPanel cron job, which uses jailshell