Im pretty sure that freezing destroys DNA in all but a few animals, and those animals are from extreme places (like 1 species of fish near the north pole, a cricket in russia, and the "water bear" which is a neat little microscopic critter that looks like its made out of jello and eats moss).
Plus the human pinworm eggs can only remain viable for 6 weeks, if they dont find a host within that time they die. and since each species is host specific there is a very small chance that you could spred them unless you are constantly touching host rats then non host rats. However, if it is the rat pinworm we are dealing with, I dont see why they are worried about the mice. pinworms are very host specific, if conditions inside the host arnt just right they cant establish themselves.
A random fact you can tell them: they themselves are hosts to nematodes... Walking barefoot outside, swimming in untreated water, touching anything in a daycare, coming into contact with any fecal matter, even at a microscopic level, etc are all ways that they could have caught parasites. You could remind them of this fact...
For more info on water bears look up Echiniscis spp.
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Leopard Geckos
African Fat Tailed Geckos
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