You don't need anything. Actually, you need food and water, but that is academic. As for "needn't": You needn't do anything. Actually, you need to work and earn money, but, once again, that is academic.

I don't need anything is correct or you can say, I need nothing

In the clause, "I do not need anything/anyone/something", a noun follows the verb "do". When we use "I needn't ...", a verb follows needn't (needn't = need not = I do not need). If you use "needn't" then the verb is in the simple infinitive form without the "to", for example, I needn't call you on Saturday because you won't be there to receive my call. for example, You needn't study anymore, the exam was yesterday!!! But "needn't" is a higher form of English, and some would say even pretentious, so it is almost never used.

You don't need anything? Is there anything you do need? You needn't be shy! :P

"Needn't" is only used to say that you aren't required to do something (e.g., "You needn't bring anything to the party, we have lots of food"). So "I do not/don't need anything" is correct.