I think there are significant differences between the Borg and zombies (although the answer to this question depends on how you imagine zombies).
I've never thought of zombies having a hive mind; they don't have much of a mind at all. Zombies don't think or talk or adapt.
The Borg do communicate and are very intelligent. Adaptation is their hallmark, of course.
As for proliferating through direct contact and puncturing, that could be said about any sexually reproducing species.