Vermin are not "tame" in the same sense as creatures, so they don't respond the same way to meeting zone designations. Private cages are the best bet, I believe.
For vermin, just build a glass terrarium. since they cannot gnaw their way out from them, they are pretty well contained.
Well, assim beat me to my usual pithy remark, so... yeah, what he said.
With rare exceptions, all material in DF is as strong as any other material, and that's 100% resistant to "breakage". A soap wall or bridge or door is as strong as an adamantine one, and a charcoal trade depot as durable as one of glass, wood, or granite.
(The exceptions are "quality" combat modifiers for weapons/armor, "fire-safe" and "magma-safe" materials, and rope vs chain "restraints" for berserk dwarfs. Also "Building destroyer" creatures - see the wiki for more info on all those.)
That will enable you to use their cage, and if you make them available for adoption, every dorf kid in your fortress will be ecstatic, lols.
The game is not intuitive, Fredd - don't be guessing, or (better) don't present a guess as a gospel answer. (I've made that mistake, most have, or seen the reaction and avoided it.)
Only the dwarfs that enjoy that vermin/animal will get a good thought - dwarfs that
dislike that animal/vermin will get a worse thought. You'd have to go through your dwarves' profiles to see who likes what to know what's safe and what's not. Private cages in bedrooms or work areas would be safe if the negative dwarves don't go there.
(Might be an obscure use for burrows, to split the population into two and have two zoos with selected negative vermin, hmmm.)