With regards to why Q described the drone as “Not a he, not a she” — in the context of the episode, this looks like Q highlighting how foreign the Borg are to the Federation.
The whole point of Q taking the Enterprise to meet the Borg was to make the point that there are life forms in the galaxy for which the crew has no adequate frame of reference.
Picard, you are about to move into areas of the galaxy containing wonders more incredible than you can possibly imagine... and terrors to freeze your soul.
Right after he describes the apparent gender-neutrality of the drone, Q hammers his point home again:
Interesting isn't it? Not a he -- not a she. Not like anything you've ever seen.
Q was attempting to narrate their reactions to this drone, rather than give a full and complete description of the Borg as a species/collective.
With regards to Borg gender in general, it seems that drones don’t exhibit much in the way of gender-specific appearance or behaviour, unless the Collective decides that special circumstances make it advantageous. Specifically:
In The Best of Both Worlds, Picard is turned not just into a Borg drone, but into Locutus — a Borg with a name. It’s not particularly relevant to gender, but this, I think, is their attempt to adapt to the Federation’s love for individuality and negotiation. Look, here you go, here’s a Borg with a name you can talk to about how we’re going to assimilate you all. You’re happy now, right?
In Star Trek: First Contact, the Borg Queen attempts to use sexual gender relations to make Picard and Data do what she wants. I think this is because humanity has proven a particularly difficult assimilation nut to crack, so they’re breaking out time travel and sexy disembodied aliens in an effort to get it done.