> Can EU laws actually override the constitutional rights of member states?

Sometimes yes.

> I was under the impression that the principle of supremacy isn't absolute and doesn't extend to overriding a country's fundamental constitutional rights.

What are a country's fundamental constitutional rights can be "dynamically adjusted" depending on the political wishes. :-(

> With authoritarian regimes gaining power everywhere, it would only take a few of them working together to pass an EU law that makes everything fair game.

There is a reason why more and more EU-skeptical movements gain traction in various EU countries.