It’s hard to evaluate the mechanical side of the situation without knowing the submodel of Integra, engine size, and whether it’s a front or rear wheel stud.
While replacing lug studs is a fairly simple procedure on some cars, that’s not the case on all cars. There are some designs where lug studs are not meant to be replaced separately, and are only provided by the car manufacturer as part of the wheel hub, often in a hub and bearing assembly. On these designs it may or may not be possible to just replace a stud, and if possible may or may not be up to standards. Thus it may be that replacing a bearing/hub assembly was the proper way to repair the broken stud.
As to whose responsibility it is to bear the cost of replacing a broken lug stud, it’s not always a simple and clear matter. Certainly if the shop failed to exercise proper care, they should be responsible. But studs are known to sometimes break even with careful workmanship, sometimes without any warning, and often without a reason that can be definitely determined. The seed may have been sown months or years ago by overtorquing, with no satisfactory way to prove that’s the case nor to positively identify who might have done it. In some states the legal responsibility may lie with whoever last had their hands on it, but in most cases it’s not possible to make a compelling case as to what caused the problem.
It’s certainly frustrating to have a new (and expensive) problem arise for no apparent reason, and naturally tempting to figure the blame lies with whoever had the part in hand when it broke. Nevertheless, it may not be the shop’s fault.