Can I substitute vinegar for eggs in a boxed cake mix?

 Because of the surprising similarities between vinegar and eggs, you mean? As in, just the other day you took a bottle of vinegar off the shelf and thought, "boy, I'd like to scramble me up a bunch of vinegar!" Or, "wow, I could sure use some poached vinegar right about now."

No. No, a thousand times, no.

Eggs provide texture, structure, richness, fat, and flavor to a cake. Vinegar would provide sourness, would make the batter runnier, and would not provide any benefit to the developing structure of the cake. Vinegar is essentially water with about 5% acid, after all.

If someone suggested you do this, stop listening to that person.

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