Will it harm stock finishes?
Eezox will not harm either wood or gunstock finishes if you simply use it only on metal, which is where it belongs for lubrication and corrosion prevention. There is absolutely no reason to put any petroleum based gun oil, synthetic or conventional, on your gun stock.
You may get some advice saying that Eezox will not harm wood finishes or the wood itself. The archives here are full, of threads concerned with de-oiling gun stocks and attempting to undo the damage caused by oil soaking. Do yourself a favor and stick with appropriate wood care products.