If you get the PSI knock off of the Beall collet chuck, which is a knock off of metal working tooling, it comes with several collets. You will want a 5/16 chuck to hold the "B" sized mandrel for sure.
1/2 inch and 3/4 inch plastic pipe from the home depot or similar allows you to make all kinds of insert mandrels (sub collets, and split sleeve friction holding adapters)-- and yes, a square piece of wood can be mounted in a collet chuck, but does not have the ability to hold as well as a round piece. -- think light cuts and avoidance of catches along with use of a tailstock.
the full range collet sets make a huge advantage if you move to metals, and especially smaller sizes. Collet technology is old stuff going back to the WW watch makers systems of the 1800s that are still running for many yet today.