I mean even if they do start using rectangular wafers for _some_ things, there is so much supply chain momentum in circular wafers that surely you have some fairly significant job security.
But I'll reiterate I mean all this on a lighter note, don't think it will happen within my career :).
Edit: Didn't read the question you asked in the first sentence. The left over parts are usually just taken along in the process till the end and are scapped when we get to the cutting stage. The problems arise because the structures on the partial dies are not the same as the full dies in the middle of the wafer. This causes a bunch of weird stresses on the edge and in my small corner of engineering we optimize the fuck out of the edge dies so their stresses are less weird.
Gather them up into a ball, then flatten them into a new wafer using a rolling pin.
They keep fine in the fridge for a few days and are still pretty good reheated with a little Tabasco sauce