A single rider could have used a buckler on the forearm of his firing hand to protect his chest and face as a default posture. Horses could have been outfitted with thick leather armor. For fights on foot, buddy teams could have alternated holding a tower shield and reloading while their partner engages the enemy by rifle.
They had some learned commanders leading the westward expansion by force of arms, who must have had an idea that protection could be helpful. But I've never heard or read anything about cowboys carrying shields. Did they? If not, why not?
Whether to use shields, armour, and horse armour is a question of effectiveness vs weight and inconvenience (and cost).
Arrow-proof shields are often about 3kg (and sometimes approach 5kg). Much heavier than this, and they're too heavy to be practical. Of course, they can be made lighter by being made smaller, but then they cover less of the body. Bucklers are useful with hand-to-hand weapons like swords, because they can be used to actively block/parry attacks, but against arrows, larger shields provide more protection. For cavalry, large shields can be cumbersome, so smaller shields are usually preferred - often about 60cm (24") in diameter. Such a shield would often weight the aforementioned 3kg. A 3kg shield on your arm would hamper the use of a gun.
Similarly, horses can be armoured, but arrow-proof horse armour with reasonable coverage might weigh 20kg.
But these weights, and the inconvenience, can be worth it, for arrow protection. What you describe with shields, and large shields for use on foot, and horse armour, is essentially what was used in American aboriginal warfare on the Northern Plains in the early 18th century. This was the time between the arrival of the horse and the gun. The horse brought with it the use of smaller cavalry shields, and for a relatively short period, horse armour. Warriors fighting on foot would use larger shields of the type in use before the horse.
But when the gun appeared on the battlefield, this armour largely disappeared (small shields continued in use, often made very thick to try to stop bullets). Large shields and horse armour were seen as worthwhile while the main threat was arrows (which these would stop). When guns (which these wouldn't stop) became a significant threat, these became undesirable extra weight, and they fell out of use.
If the only guns in the Indian Wars had been on the US side, then shields, armour, and horse armour might have been quite worthwhile. However, both sides had guns. The armoured cowboy would have been shot by a musket.
For a description of 18th century warfare on the Northern Plains, and the changes in response to guns, see chapter 10 in Bohr, Roland, Gifts from the Thunder Beings: Indigenous Archery and European Firearms in the Northern Plains and Central Subarctic, 1670-1870, UNP - Nebraska, 2014.