Do we know exactly what the twelve stones were?

Not all of them. The WEB names them: "a row of ruby, topaz, and beryl" (28:17), then "a turquoise, a sapphire, and an emerald" (28:18), "a jacinth, an agate, and an amethyst" (28:19), and "a chrysolite, an onyx, and a jasper" (28:20). The Hebrew names are harder to pin down. Brown-Driver-Briggs calls the onyx, shoham, "a gem, identif. dubious". Strong's calls the Hebrew word behind the WEB's turquoise "a gem, probably the garnet", and the one behind jacinth "a gem, perhaps the jacinth". For ten of the twelve, Strong's entry says probably or perhaps. What the text is sure about is what was engraved on them: "The stones shall be according to the names of the children of Israel, twelve, according to their names" (28:21).