Will a campagnolo square spindled BB fit a Shimano square spindled crank?

Yes and no. There are two similar but not identical square taper designs, ISO (Campy and a few others) and JIS (Shimano and most other Japanese makers). They share the same 4° included taper angle but the ISO version is slightly smaller across the flats at the ends. So, in theory, a Shimano crank will slide too far up an ISO taper and a Campy crank will not go far enough up a JIS taper.

That said, tolerances being what they are, using them interchangably does work sometimes. Here is Sheldon Brown's very detailed article on these spindles:

https://www.sheldonbrown.com/bbtaper.html