Having worked as a lonf distance lorry driver, I can sympathise a little, I too hated nights away, I did the sensibe thing and got a job that didn't involve them.
This is a tough one to call for you. As you have no contract, from a legal standpoint there is no compulsion on you to work away. A lot of employers have this "at any company site" clause in their contracts, but as you don't have one then there is nothing to enforce is there??
That said, from a plant earth standpoint, I think you can be assured that if you don't you will be walking quite soon. There is a balance to be struck, as if you give in straight away I can guarantee you that your name will be first on the list next time. The trick is to come to the table with a ready solution, such as, "if we have to work away, then let's have a rota for everyone to do it" As soon as you are told that so and so cannot do it for whatever reason, then that becomes your reason. Simple.
That said, it might still cause difficulty, my advice, take it on the chin, while you look for a new job!