When a He atom is ionized, which electron is removed first, the one with the spin up or down? or is it just random?
An interesting question! Does the polarization of ionizing radiation interact with electron spin? Apparently, that has been investigated to some extent, and some spin information can be obtained from an entangled electron and photon.
So the answer to your question is not a clear-cut "yes" or "no", but depends on the parameters of the incident photon.