Compounds that adopt the zincblende structure Zinc sulfide, copper iodide and copper bromide all adopt the zincblende structure. My lecture notes suggest that this structure is adopted preferentially over the wurtzite structure if the bonding is more covalent (why is this?); thus, it’s adopted where there is a polarizing cation and a polarizable anion. 

However, surely a polarizing cation should be small and highly charged (i.e. not ions like $\ce{Cu+}$ surely? It has a low charge and large-ish radius). What makes this polarizing? I’ve also heard this about $\ce{Ag+}$.
Generally when you get a preference between two structures due to directional bonding, the first order environment of the cation is different. In this case however both are tetrahedrally coordinated. The Wurtzite structure has a marginally greater Madelung constant (1.641 versus 1.638).

Of the top of my head, I think the dominant factor in favour of the blende structure is stronger directional cation-cation and anion-anion interactions. In the case of Cu(I), the d-orbitals are going to be fairly diffuse due to the low charge, potentially allowing some covalent interaction between adjacent sites. A much better example would be GaAs, where both cation and anion are realtively soft/polarizable, which adopts blende, in contrast to AlN, with hard ionic interactions, which adopts wurzite. I don't think it's really as simple as this because the difference in energy between the two structures is often very small; you can see this in the relative stability of both ZnS polymorphs.