Learn Farsi: آنها برای رهایی، هیچ‌چیز جز نخوردن گوشت را انتخاب نمی‌کنند. - For liberation, they choose nothing except not eating meat.

You take the standard infinitive خوردن (to eat) and add the negative prefix نـ, making نخوردن. Because it is used here as a verbal noun, it translates to the concept or act of "not eating".

How are the words نخوردن and گوشت connected?

They are linked by an invisible Ezafe: نخوردنِ گوشت (nakhordan-e gusht). When a verbal noun takes an object, the two are joined by an Ezafe. You can think of it literally as "the not-eating of meat".

Why is the object marker را placed after گوشت instead of right after هیچ‌چیز?

The marker را applies to the entire direct object phrase. Here, the complete object they are acting upon is هیچ‌چیز جز نخوردن گوشت (nothing except not eating meat). Therefore, را anchors to the very end of that whole block.

The English translation says "choose nothing," but the Farsi verb انتخاب نمی‌کنند is negative. Why?

Farsi requires double negatives. When a sentence contains a negative word like هیچ‌چیز (nothing), the main verb must also be made negative. Literally, the sentence says "they do not choose nothing".