Not entirely sure, but I think I can discern a seated left-facing Britannia in that first image. "Slightly larger than a quarter," as you say, would probably make it a British half penny or maybe a lookalike Colonial or other privately-minted token, which circulated as commonly or even more so at times than the real thing.
Since the coin's edge has no discernable raised flange, and Victoria's half pennies' Britannia faced right, my hunch would be that it's older than the "1879 or 1878" you mention.