Loretta,

My honest answer is that I do not know. I have seen HUD 1 forms on which individuals are specifically identified by name for the services they perform, and payment is generated to them. I have accepted work from clients, and seen my name printed on the HUD 1. 

However, if you are going to say no to a questionable company, it is worth a try. If they give you an answer like that or any other reason for not putting you on the HUD 1, you have lost nothing by telling them to take a hike. You were going to turn them down anyway because of their payment history. Remember, you would be doing this only with problem clients...not the clients that have demonstrated a good faith payment history.