can you get fired from a job if you do not want to give your social security # for clearance to another site?

we have a huge project that was given to us, now the agency needs ss# from the drivers whom are delivering the merchandise for clearance. My driver refuse to give his ss# because he says its personal and he is afraid of identity theft can we fire him because of this.....? also can the employee file a suit with the dept of labor because he was fired