To be clear, what I'm saying here is that this claim maps into our model of neurology as the equivalent of a claim like "the earth isn't a triangle." Whether or not it's supported by any evidence, what it actually is is a positively-phrased restatement of part of our set of axiomatic priors.
We know that the Earth can only be one shape; we know there are an infinite variety of shapes that something can be; and so our priors contain a set of all the claims like "the Earth is potato-shaped" and "the Earth is a doughnut" all with extremely low probability, before we encounter any evidence at all, just because the probability-mass has to get spread out among all those infinitely-many claims.
Assuming continued lack of evidence either way, a claim like "the Earth is not [one particular shape]", then, doesn't require argumentative support to be taken as a default assumption (as you might in e.g. the opening of a journal paper.) The probability of it being any particular shape started very low, and we've never encountered any evidence to raise that probability, so it's stayed very low.
(Yes, that even applies to the specific claim that "the Earth is not an oblate spheroid." If we never encountered any evidence to suggest that claim, then it'd have just as low a default confidence as any of the other claims it competes with.)
For claims with no evidence either for or against them, the analytical priors derived from the facts about the classes of claims to which the claim belongs, determine where the burden of proof lays. Low-confidence priors? Burden to prove. High-confidence priors? Burden to disprove.
In this case, we already know that neurons can do several things, and AFAIK we've never encountered any evidence of neurons having specialized functionality, or any evidence that neurons don't have specialized functionality. Our tools just aren't up to telling us whether they do or not. But, because one claim actually factors out to several claims (neuron specialization → lots of different ways neurons could specialize) while the other claim doesn't (neuron generality → just neuron generality), the probability-mass ends up on the neuron-generality side. (This is another way to state Occam's Razor.)
Mind you, this might be entirely down to our inability to study neuronal dynamics in vivo in fine-enough detail. In this case, our lack of evidence doesn't imply a lack of facts to be found, because we have no evidence for or against this hypothesis. Instead, it just determines what our model should be in the absence of such evidence, until such time as we can gather evidence that does directly prove or disprove the specific hypothesis.
Or, to put that another way: if humans only ever studied bees from a distance, the default hypothesis should be that all bees do all bee jobs. The burden of proof is on the claim that bees specialize. Later, when we get up-close to a beehive, we'd learn that bees do specialize. But that doesn't mean that we were incorrect to believe the opposite before. Both our belief before the evidence, and our belief after the evidence, were the "correct" belief given our knowledge.