I am such a newby that I don’t have forge, anvil or tools yet but have strong desire. I’ve been studying. 
In one project design it was stated that if only riveting in one place a round rivet would allow movement where a square hole and rivet will not. I find no information on making or using square rivets. Is this unusual?

To make a square rivet, just start with square bar stock rather than round.

Round rivets are most common in things that are meant to turn, i.e. tongs.  Round rivets are used for fastening as well, some structural work, machinery, ornamental work, etc.

The square rivets would be used where any movement would not be wanted such as joinery work.  You would forge out the rivet stock in a square shape if you started with round stock, punch a properly aligned and sized hole with a square punch and rivet the pieces together.  The rivet head could be shaped either square or round.

You can always put a round rivet into a square hole and as you form the head it will fill most of the square hole as it collapses. For this you need to heat the rivet in the forge, don’t just use a torch and heat only the side you riveting. Also, you will need to start with a longer length than the standard 1-1/2 ~ 2 time the diameter. This won’t give perfect results, but perfect is not always needed.

To prevent rotating, you can also put chisels cuts into the hole that the rivet is formed into. It comes down to how much rotation resistance you need.

Keep it fun,

David

Yes, a aquare rivet will resist rotational forces if that is needed in a particular application.  However, while you can punch square (or any other shape) holes often times it will be easier and faster to install 2 round rivets than 1 square rivet to resist any turning motion.

"By hammer and hand all arts do stand."

Also, it can be hard to align a punch well enough that the finished piece looks straight.  And all things being equal, a square rivet will withstand a lot less torque than two round ones (or two square ones for that matter).

But if you don't want your rosette twisting under idle fingers, a square rivet can be a pretty slick choice.