Eveready Gas? Eveready Oil? What of Eveready batteries? I don't know if it's the same company; the wikipedia page for the battery firm says nothing about a sideline in oil.
However: I found a page on ebay that sold a gallon on Eveready-branded Prestone, and that's a brand that was owned by Union Carbide. Which owned Eveready batteries. So yes, I'd say it's the same company.