Sounds like a fun project! Great questions about the braces and layout system.
I find that barns built in the last half of the 19th century in Ohio are as likely to have pegged as unpegged braces. Although intriguing, I'm not sure the "wiggle" theory matches what the barn framers had in mind in frames with unpegged braces. My own engineering background temps me to believe that the carpenters were well aware that braces are meant to work only when they are being pushed on, or "in compression". That being the case, pegs were not needed, and in fact could work against the frame in certain conditions.
There are several reasons the unpegged braces get loose. Shrinkage and movement of timbers are likely most of the cause. The timbers shrink in thickness which may lengthen the brace "leg", and timbers can bow or twist when drying which will also undo the brace layout. Foundation subsidence and frame modifications are also often culprets.
I would strongly recommend against screwing up your braces (I couldn't resist). Although Merle Adams taunts us to "confuse" the load paths in our frames, he is I assure you, speaking in jest. By attaching the braces in place with screws, the force of compression is transferred to the screw rather than the shoulders and brace tenon. Depending on the orientation of the screw, the problem may solve itself quickly by the screw shearing off. If the screw can handle the load without shearing, the load will be transferred to the screw's anchor point. This can cause distorted and eccentric load paths. It can also cause shear if the brace goes into tension when the frame "wiggles".
I would recommend simply "wedging" or "shimming" the braces that are loose. I moved an 1895 barn with unpegged braces in 1994 to our property and drove wedges in at the bottom of the barces that were loose. The barn is very tall and absorbs a lot of wind. The braces remain tight today.
As far as whether your barn is square rule, it's very likely it is since scibe rule had been largely replaced about two generations of barn builders prior to the framing of yours. It's also possible with that date it is "mill rule", but not if it's hewn. The simplest way to identify square rule is to look at the joints. If the only wood removed at the mortice is for the tenon at nearly all of the joinery it's probably scribe rule. If so, there will be matching carpenters marks that look like roman numerals at the joints. If however there is an area cut into the timbers around the mortices, that accomodates the timber as well as the tenon, it's probably a square rule frame.
Please feel free to ask questions as needed to fill out your survey for TTRAG. It will be great to see a Wisconson frame in the archives! Keep up the good work.