I prefer that he (should) not go to USA.

    此句正确。如：I prefer that you (should) wait here.我希望你在此等候。（摘自《英汉多功能词典》新编升级版）

    I prefer that he didn't go to USA.

    此句错误。如：In the past men generally preferred that their wives _____

in the home. A. worked  B. would work  C. work   D. were working  （全国研考，92-5，答案是 C）

    I would prefer that he didn't go to USA.

    此句正确。如：Val would presumably prefer that you didn't get arrested. 

    I would prefer that he (should) not go to USA.

    此句正确。