The situation you describe does not exist because the government isn’t controlling my most basic rights enough. It’s because those who supposedly represent us are doing everything they have no legitimate power to do, and few of the things they’ve actually sworn to do.
I agree completely with the second sentence and I think we'd agree that much care must be taken to not go further down the road of the fedgov arrogating more unconstitutional power to itself. I'm still struggling to get a grip, though, on the crux of your e-Verify objection. Is it that you fear an e-Verify database would morph into a general ID required for circumstances other than proof of citizenship?