Learn Dutch: Ik bestel de garnaal nooit en de champignon evenmin. - I never order the shrimp and neither the mushroom.

Evenmin is an adverb meaning "neither" or "nor". The second half of this sentence is a shortened version of en ik bestel de champignon evenmin. Because evenmin acts like a negating adverb here, it sits at the end of the clause, just as niet often would.

Could I use ook niet instead of evenmin?

Yes, you could completely replace it and say en de champignon ook niet. Evenmin is a slightly more formal but very concise way to express "neither" when adding a second negative statement to a first one (triggered here by nooit).

Is it grammatically correct to leave out the verb in the second half of the sentence?

Yes! This is a very natural way to speak. Because the subject and verb (ik bestel) are the exact same as in the first clause, you can just drop them. If you wrote it out fully, it would be en ik bestel de champignon evenmin.

Why use the singular de garnaal and de champignon instead of the plural forms?

When referring to items on a restaurant menu, Dutch speakers often use the singular definite noun to refer to the dish as a whole. It essentially means "the shrimp dish" and "the mushroom dish".