Right, but isn't it a problem that a quine also requires the information contained in the language's compiler/interpreter to be fully meaningful? This would be "outside the universe" so to speak.
It really doesn't matter what Turing-complete language you choose; they can all be implemented in terms of each other, so as soon as you have your quine one language you could as well do it any other.
An assembly quine is an odd thing to even ponder.
Ultimately, if you can't write a Quine directly in logic gates, which you can't because no microprocessor can output another microprocessor, you need something external to the "universe" of the language.
You only need an interpreter for that language.