UPDATE: Elliot Glazer outlined a proof of nonconservativity of Z + GC over Z in his answer (on this page) to this MO question back in November of 2021; he has now (Dec 20, 2023) posted a paper on arXiv that fleshes out the nitty-gritty details of the argument (see here).
The known proofs of conservativity of ZF + GC (Global Choice) over ZFC make significant use of replacement, and as far as I know the problem of conservativity of Z + GC over ZC is wide open.
Let me add that I have discussed the problem with a number of experts over the past two decades, and also posed it on FOM in 2006 in this posting.