Both are fine. You could also ask the second one like this- You don't want to know either?

Only the first sentence sounds right. Either is not used the way it is used here. The meaning of "either" that you want to use is "also; too; as well; to the same degree". Either is used as an adverb only after negative clauses coordinated by 'and', 'or', 'nor', or after negative subordinate clauses e.g. He's not fond of parties, and I'm not either. If you don't come, she won't come either. And a better way to say the same thing is : Wouldn't you like to know too ? Won't you be interested in knowing as well ?

in English, it's best not to repeat the same word in the same sentence. just saying, there are many other words that would fit better than "don't" also if this was professional, try to avoid using contractions Wouldn't you like to know as well? Wouldn't you want to know?