[RWebb]

some species (and genera) of the things called stinkbugs are indeed Asian, BUT others are native to North America

actually, your logic is flawed - a zapper will only kill bugs in the immediate area; then the area will be recolonized by bugs from nearby - the best recolonizer will be there the "firstest, with the mostest" of its friends -- if that is the predator (unlikely) then great; more likely it will be the bug your tried to get rid of (prey)

you can always try a zapper and see what happens - it is late in the year, so they will all be gone soon anyway in most places

IIRC, there are 2 species of exotics in different genera - they do indeed cause plant damage - they pierce the plant tissue and suck out its "juice"

the exotics and the natives are becoming more & more of a problem due to global warming - it allows them to have more generations per season in much of their range

also, I think these really are bugs (i.e. Hemipterans)