Unfortunately this is just a tricky part of Czech but you might want to try to think about it as the negative statements tending to just build up and 'confirm' each other. German and English tend to look at these statements as a math problem so Czech is tricky... especially because sometimes it works like we're used to.
As others have been pointing out, this construction is very odd in English.
To ask 1) "Was he ever sick?" means the speaker does not know - he could have been sick zero times or many times.
2) "Was he never sick?" responds to a different context. The person asking might already have guessed or learned that the person has never been sick, and this question is asked to verify, emphasize, or repeat in surprise that the person was never sick.
If there is just this one construction in Czech, and context creates the appropriate emphasis, then I think "Was he ever sick?" is the better translation in English.
But if this construction in Czech is emphasizing that the person asking already knew/suspected that "he was never sick", then how would one construct the more common form in English for "Was he ever sick?"