No, a monohull sailing along under jib alone is not a unirig under any interpretation of the term. It is a sloop rig temporarily sailing under jib alone.

There are a few, very rare, boats that do not have a mainsail and only have a headsail. But now that I think about it, even those usually have more than one headsail, so those would not be unirigs either. So that leaves us with the kitesail boat, if the kite is the ONLY sail, to qualify as a unirig when no mainsail is involved.