Why can't we say [They have never cooked A crab]? It sounds more natural in English.
Using "crab" without the article indicates that your talking about a general type of food.
They've never cooked crab; They've never cooked chicken; They've never cooked beef.
Sounds perfectly fine to me (native English speaker).
To say "They have never cooked a crab." in Dutch, you would also need the article.
"Zij hebben nooit een krab gekookt."
Honestly, I don't think "They have never cooked a crab" sounds any more natural in English than "They have never cooked crab."
If anything, I tend to think the opposite is true, but this may be a regional issue.
I think only a tiny number of English speakers would even notice the difference.
Well, I was always thought, that when there is a negation in the sentence you should use "A/AN". You would rather say "I have never ridden A bike" instead of "I have never ridden bike". Kai_E is right about the issue with type of food, because one would say "I have never cooked rice". But for me "a crab" is not a type of food, but a particular animal. That is why "I have never cooked A crab" is in my opinion correct.
@sobmar, Rice is uncountable, that's why it never gets an indefinite article.
I hesitated whether to put "a" here or not,following the structure of the Dutch sentence, and decided to insert an article as it sounds more natural to me too. And it was marked as wrong [sigh]. But I'm not a native speaker, I may be mistaken.