My question here is that can we use never plus did plus the first form of verb?
Grammatically, yes, "I never did say that" is fine, but unusual, and wild only be used for some specific reason.
Reasons you would choose it over the simpler "I never said that" might be if you wanted to emphasise that you choose not to ("In the end, I never did say that."), or if you want to contradict something ("You said the Earth was flat!" "I never did say that!"). "I never did say" places a lot of emphasis on never did, and you would need some reason for that emphasis.