Here is a discussion on the topic. https://www.italki.com/discussion/216919

I think that, while they both mean roughly the same thing there is a slight difference. If you say "I didn't buy eggs because we already had some" you're saying you didn't buy eggs. "Did you buy eggs? I didn't buy them because we have some". If you had gone and bought some eggs, then realised you already had some you would say "I needn't have bought eggs...". Meaning I bought eggs, but I shouldn't have because I didn't need to (because I already had some). Or that's how it reads to me.

As jennifer mentioned, the commonly used one is definitely: didn't need to buy.

Both can be used. In my experience (as a native speaker of British English) it seems like "needn't have" is the least used of the two and when I do hear it it's usually an older person saying it but both are fine.