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.