I still don't think it's right, though. Firstly we have a problem with defining subjective/objective. For example a decent hammer is a high quality tool for hammering in nails, but a low quality tool for screwing in screws. Does that make the quality of a hammer subjective (depending on the context in which it is used) or objective (independent of human perception provided we can accept an objective goal of joining two pieces of wood) or subjective again (because it takes a human to decide on that goal)? I find it near meaningless to begin this debate without agreeing on terms.
Second, going all the way down to electrons, protons etc - this liking and disliking seems to be confusing different types of process altogether: computing/selecting from possible alternatives vs simple electromagnetism. That's not to deny that our brains might be powered by simple electromagnetism under the hood. But if you define "liking" for an electron as "moving towards a positive charge (momentum and other forces/quantum effects permitting)" and "disliking" as "moving towards a negative charge (similarly qualified)" then the electron will never experience the sensation of "dislike", while we as humans often do, so the parallel seems inappropriate.
As to whether quality is objective/subjective what's wrong with the explanation that individual tastes vary but there is usually some correlation between them due to the evolutionary forces that shape both us and our memes?
As to whether like/dislike defines subject/object what do you make of the person who dislikes their own body and would rather inhabit a different one?
As to whether quality is essential for definition of subject/object: I'm not convinced it is at all. Sure "quality" might break a crude model of subjectivity/objectivity (which can then be refined as above) but just because something broke our model doesn't make that thing essential to defining the model. I can break some form of logic (propositional? first order? I forget) by asking you to evaluate the truth of "this statement is false" but that didn't mean we needed that statement to define the logic in the first place.
And as to whether quality is undefinable: just because we repeatedly fail to define it doesn't mean it can't be.
Not wanting to be argumentative btw. I'm genuinely interested to get to the bottom of this if you have further thoughts.