I am certain the outside material you claim is white and clear is a silicate, either quartz or chalcedony. Many rocks dragged around by glacial and alluvial flow are cemented together by a silicate enriched solution in the waters of glacial melt. It cements together and forms rocks like puddingstone. If you don't already know quartz and chalcedony are identical. They are chemically SiO2. The difference between then is quartz is macrocrystalline and chalcedony is microcrystalline. Macro is larger crystals. Micro is smaller crystals
The crystal formation is due to different conditions during the formation. The two rocks feel differently. Quartz is coarser, you can feel the grain. Chalcedony is smooth and has a waxy feel. Quartz will have cracks in it but chalcedony has concave divots like the ones in flint tools. Those are called conchoidal fractures. It usually does not have fissures. Agates are layered chalcedony. They form in volcanic rock which is porous. Over time, silica rich waters flow into the porous rock and settle in voids caused by air bubbles in the rapidly cooling lava. Over the years the water flows in, evaporates and leaves a layer of silica with varied trace amounts of minerals giving them colors. Alot of red and white in the Lake Superior region.
As for the unusual rock inside. This is conjecture. Petrified wood is actually in many cases replaced by silicates as the silica solution soaks into the wood and the wood breaks down. It is actually agatized. Something like that could be picked up be glacial flow and deposited anywhere by advancing and retreating glacier. Then, the outer coating is formed by alluvial flow from the retreating glaciers. The river valleys are geological depositories from glaciers, floods and erosion. My opinion. Not written in stone. Lol.