In a nutshell, the DNA fails at protecting from changes all the time, and the immune system is constantly checking for that and killing off those cells.
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.
But I may have been a bit unclear about my mapping of computer and biology terminology to this question.
I tend to think of a computer programs as being mathematical machines, and proteins as being molecular/chemical machines. Motor proteins "walking" along microtubules is the most vivid example of this.
I'm sort of torn between both worlds here, honestly. I like both perspectives. Viruses especially have a very quinish quality to them in that they're stripped down to just enough stuff to get a host cell to make more viruses.
Or if you put a fully-functional cell in the vacuum of space, or the heart of a star, it won't replicate either.
All replicators need a suitable environment to replicate in. The fact that inhospitable/non-viable environments exist does not negate the fact that they are actually replicators.
Symbiotic replicators that rely on the presence of each other to reproduce (like cycles of quines?) are also still replicators.
The DDC model[0] proposes that one of the drivers of evolutionary change is polyploidy, where after an organism has multiple copies of the genome, genes are free to drift into new functions.
An organism with four copies of a gene instead of the usual two is more rad-hardened than the basal non-ploid organism.
[0]: https://www.researchgate.net/figure/The-duplication-degenera...