To see where antireductionism has value consider the natural numbers (that is, positive integers) where one does not need antireductionism because of the presence of unique factorization. Given a natural number there exists, although it may be impossible to find within the finite age of the universe, a unique analysis of that number into its prime factors. That analysis reduces the natural number to those factors. The synthesis occurs through multiplication which unambiguously brings us right back to the original number.
If reality were so easy to analyze and synthesize as factoring a natural number, that is, if reductionism were true, there would be little doubt that the reality that we are part of would be just as deterministic as mathematics.
The antireductionist position has epistemological and ontological components. The epistemological component is like the problem of not being able, within the lifetime of the universe, of actually factoring most natural numbers. The ontological component is totally unlike the situation with the natural numbers. That claims that we cannot reduce everything to something else, say a quantum reality. It is like saying that there exist some natural numbers that we cannot, even in principle, factor.
Is there anything in reality that might suggest ontological antireductionism is true? The article suggests one can find examples supporting antireductionism in areas such as cybernetics, systems theory, history and economics.
Now to answer paraphrases of the questions:
Can antireductionism be a scientific position?
To say antireductionism is not a scientific position would be to say there exists a demarcation between what is science and what isn’t. Perhaps antireductionism is a kind of pseudoscience? One of the proponents of finding a demarcation to separate science from pseudoscience is Karl Popper. According to the antireductionism wikipedia article, Popper “was a famous proponent of antireductionism” in his book Of Clouds and Clocks: An Approach to the Problem of Rationality and the Freedom of Man. So, not all ways to demarcate the boundary between science and pseudoscience reject antireductionism.
If everything is made of quantum particles can’t we use this one framework to explain everything?
A reductionist might answer yes. An antireductionist would more likely answer no. If the reductionist is correct, reality may be like the natural numbers. We are part of reality, so there would be a reductionist explanation for everything we do. That would suggest superdeterminism were true and everything is determined. That would include, according to John Bell, “the 'decision' by the experimenter to carry out one set of measurements rather than another”. So if that reductionist view is correct the reductionist needs to find a way to avoid superdeterminism or give up science itself.
There is a comment asking "What is eliminativism?"
A reductionist position attempts to analyze something into its components and hopes that such analysis can replace what one started with. For example, 6 can be analyzed into the factors 2 and 3. An eliminativist position would say that there is no need to bother reducing something. It is illusory. Succeeding at such an argument is to say that something does not exist. It would be like saying we do not have to bother factoring the imaginary number i in the natural numbers because i does not exist in the natural numbers. Believing it does is an illusion.