Can you play Star-Crossed even with no friendly units on your side to return an enemy unit to the opponent's hand?

No, you cannot play Star-Crossed if you have no friendly units on the board.

As established by RiftJudge FAQ #9772, while you do not need to "own" a unit (in the sense of having it in your deck), you must control a friendly unit on the board to serve as a valid target for the spell. Star-Crossed specifically 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 to target, you cannot fulfill the requirements of the spell, and therefore you cannot play it.