I'm not sure how you would disprove the hygiene hypothesis, because it is a really weak claim and rejecting weak claims is really difficult.

The anti vaccine position makes a very strong claim, namely that vaccines will cause complications that are strong enough to justify not vaccinating children, which is obviously false since a lot of the diseases that are vaccinated against have actually killed children and the vaccines have dropped child mortality significantly and the complications that are supposed to be avoided by refraining from vaccines tend to be both rare and non life threatening.

You can't make the same argument with the hygiene hypothesis, because the claim is really weak. Nobody is saying that extreme hygiene will kill you. The argument is along the lines of "lack of exposure to environmental microbes, viruses or allergens may lead to an unprepared immune system that hasn't developed a wide variety of anti bodies or is more likely to develop allergies or autoimmune problems".

I'm not sure how I would be able to argue against this claim since it only takes one microbe, virus or allergen to make it true.

The context here isn't hand washing vs not hand washing, it's aggressive ozone + UV sterilisation vs regular hygiene.

Not to mention that the hygiene hypothesis has an even weaker version still, namely the "old friends hypothesis". It seems pretty weird to equivocate this to being against vaccines.