I tried to play Overclock with Stabilizer in play. I expected it to enter play, but have no effect. In practice, it never appeared in the global enchantment area.
Acmar's Stabilizer was neutral at the time, because of Drake Lair (not relevant in theory, but might have had an effect).

Incidentally, is there any way to tell if Stabilizer is on or not? I didn't see any token symbol on the card's blow-up.

There should be tokens. Probably a bug.