This is a simple matter of ergonomics. During the Civil War, the vast majority of soldiers used muzzle loading muskets. Muskets are quite long, generally somewhere around 4.5-5 feet long. In order to load a musket, you need to use a ramming rod to pack down the shot. Simply put, you can't easily ram that down while kneeling. I've heard of a technique for loading while prone, where you brace the butt of the musket against your foot while you ram the powder down, but this method had a number of disadvantages that made its usage relatively rare.

1. This is just plain harder than loading while standing up. A soldier in battle is hardly in a controlled environment, and there's all sorts of things that can go wrong if the soldier gets flustered. He might let his foot slip while ramming, he might not get enough powder to the base of the musket, he might accidentally fire while flipping to and fro. Even in the best of cases, he's going to fire much slower than a soldier standing up. This is only exacerbated if you're dealing with a bunch of inexperienced volunteers and conscripts, which a Civil War officer would be. These soldiers had months or weeks of training, which doesn't give them time to learn these sorts of advanced techniques.

2. The soldier will have a much harder time using his bayonet, either defensively or offensively. Modern man generally underestimates the power of the bayonet. If you want to defend against a cavalry charge, take an enemy position, or exploit an enemy that seems to be wavering in his resolve, there is no better tool than the bayonet. And no one has ever successfully stabbed someone lying down.