I guess the right question to ask about DNA is, does it contain all the instructions required to produce a copy of the DNA itself? DNA translation involves a bunch of RNA, sure. But DNA copying is mostly done by DNA polymerases, which are typical peptide based proteins encoded in the DNA itself, right?

So in a real sense, the DNA contains the blueprint for the machinery to copy the DNA itself, but not the instructions for building that machine from the blueprint, the same way a software quine doesn't necessarily include the source code for a compiler.

Definitely a definitional problem, but that's my shot at resolving it a bit.

Then again, DNA polymerase is not perfect, and DNA is not immune to single nucleotide changes having a real effect, so it's only a quine in the "spherical cow" way of looking at it anyway. But it's certainly the most "quinish" thing in nature that I can think of.