apparently means "I only eat vegetables"? Shouldn't there be a negative connotation with the ません-ending?
If that's the case what does 私は野菜しか食べます mean? Is there a special contextual rule with exceptions or just the term しか?
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There are good linked questions, but I think I'll try my hand at an answer.
There exist constructs and words in Japanese that either take on a different meaning, or can only be used with a negative sentence. しか is one such construct. It can only be used with a negative sentence. It is grammatically incorrect with a positive sentence.
For instance, ほとんど can be used in a positive sentence to mean "almost, nearly", but in a negative sentence, it means "barely".
誰も and 何も are other examples of expressions that can only be used in negative sentences.
In this case, you could think of it as "I don't eat anything except vegetables".