You don't indicate whether you've got a square taper (older) Record BB or one of the Campagnolo outboard bearing systems (Ultra Torque or Power Torque).
If it's an outboard Campy BB and crank, the spindle is a larger diameter than Shimano (25mm as opposed to 24mm), among probably other issues, so it will not work.
If it's square taper, Campy uses a different taper geometry at the taper called ISO, as opposed to the JIS standard that almost all other cranks use. More info on this here. As that page indicates, while there is some risk associated with having less spindle engagement, a lot of people successfully mix and match JIS BBs onto ISO cranks without issues. Ideally you get a spindle that's about 9mm shorter than whatever ISO spindle gave you correct chainline before, because the cranks each sit on the spindle about 4.5mm out from before. Note that the good way of figuring this out is before taking anything apart, first observe what chainline the old BB is giving you and whether there would be any issues with making it better if it's not already perfect, then pull the cranks and see what spindle length you've got.
Your Gios may have an Italian BB shell. This isn't a big issue but if so you'll likely need to order the new BB. Usually the BB is marked somewhere with its thread pitch, 36mm x 24 tpi for Italian.