I wonder, whether it is correct to say, that "needn't (do)" tends to be a traditional way of saying, while "do not need (to do)" tends to be like American English. [Native speakers, please kindly rectify it should the above presumption be wrong.]
Nevertheless, I personally would simply use "do not need (sth. or inf.)" to express various situations. For instant,
I need a car. (affirmative) / I do not need any car. (negative)
I need to write a letter. (affirmative) / I do not need to write any letter. (negative)
Good day and ciao.