We don't need eggs.

The sentence is a negative statement about needing eggs. 'Eggs' are countable. In negative sentences with countable nouns, 'any' is typically used to indicate an unspecified quantity. 'Much' is used for uncountable nouns, and 'some' is usually used in affirmative sentences or offers/requests. Thus, 'any' is the most appropriate choice. The completed sentence is 'We don't need any eggs.'.