Does Star Cross require you to own a unit to play it?

No, you do not need to own a unit to play Star-Crossed.

According to RiftJudge FAQ #7532, you can play a card even if you do not have a unit, as players will "do as much as they can." However, there is an important nuance regarding targeting:

As established by RiftJudge FAQ #9452, Star-Crossed requires both a friendly unit and an enemy unit to be present as valid targets to be played. If you do not have a friendly unit on the board, you cannot fulfill the targeting requirements of the spell, and therefore you cannot play it.

In summary: You do not need to own a unit (in the sense of having it in your deck/collection), but you must control a friendly unit on the board to serve as a valid target for the spell.