The squared shield in question is generally referred to as the scutum, although in Latin, scutum simply means “shield.” The scutum was carried by most legionaries and many auxiliary infantry throughout the Principate, and appears to have fallen out of use in favor of ovular shields sometime during the mid third century A.D., a period characterized by changes in the panoply of Roman infantry.

The scutum was constructed from planks of wood, assembled longitudinally and held together by glue. It was covered on both sides by linen or hide, as has been confirmed by the scutum recovered from Dura Europos in Syria. This linen or hide cover was typically painted, with designs ranging in elaborateness. In Late Antiquity, shield imagery reflected unit identity, a tradition which had it’s origins in the Principate. Further protection came from bindings, edgings of hide or bronze attached to the circumference of the shield. The scutum was carried by a single grip, protected by an iron boss (umbo), fixed to the front of the shield by a number of rivets. Although the exact size of the scutum is unclear, the scutum recovered from Dura Europos provides a generalization. Measuring three and a half feet high and sixteen inches wide, it is likely most scuta were around this size. It is worth noting that Roman shields from later periods were constructed similarly, as were the shields of their Germanic adversaries.

The size of the scutum would have provided comprehensive protection to a Roman soldier, especially when combined with armor and helmet. A scutum would thus have provided protection from sword and spear thrusts, and the use of formations such as the testudo and fulcum (in Late Antiquity) implies Roman shields were relatively impervious to ranged missile fire. Roman infantry (especially in the Principate, but throughout Late Antiquity as well) fought in close-order, and required shields that could sustain the shock of hand-to-hand combat. That said, our literary sources are silent on the “durability” of a shield, as they are in general unconcerned with technical details. Experimental archaeology offers much in our attempts to understand the practical effects of Roman equipment, yet even here we must be skeptical. We cannot truly replicate equipment such as shields in the exact manner as the Romans, nor can controlled tests recreate the combat situations such equipment saw, due in no small part to our uncertainty of the mechanics of Roman battle. Thus, while the scutum and other Roman shields were quite effective and reliable, it is unclear exactly how much “damage” a shield could have suffered whilst still remaining useable.

Some surviving examples of Scuta here.

http://artgallery.yale.edu/collections/objects/5959

https://www.pinterest.com/schollespqr/schild-scutum/

Typically the main bodies are made of wooden planks glued together. If I'm not wrong, some were made of plywood. They are then layered with rawhide and painted. The boss (umbo) and trimming were made of metal, either iron or bronze.

They were definitely effective against swords and arrows. Piercing through the shield with either would be possible but not something you'd probably count on doing (the rawhide alone is tough). Of course with sustained use like most shields, they can be damaged beyond effective use.

EDIT: Linen! I wasn't completely sure on the specifics of layering though I suspected linen was used (as with later Viking era shields) so I left that detail out and only mentioned rawhide.