Hello,
I am a reenactor and an avid Civil War amateur historian and aficianado. Carrying muzzle down is most certainly not in the regulations. In fact, "at the route step" allows the soldiers to carry their muskets anyway they desire and however it's comfortable as long as it's not muzzle down. It's expressly prohibited.
Soldiers were issued tompions (wooden plugs) to put in their barrels in case of rain or in the case of marching in the rain the order "secure arms" could be given where the musket is placed under your left arm, lock under your arm pit with the rifle upside down, that is the hammer pointed towards the ground, to keep it dry and the barrel pointed downward at about a 45 degree angle to keep rain out. That's as close to muzzle down as would be allowed.
All that being said, I'm sure soldiers did it anyway if it was comfortable and if their officers and NCOs were lax enough to allow it as many in all volunteer units were as long as you weren't loaded. Not saying it was common or encouraged although I'm sure somewhere and in different units soldiers did it even being against regulation.
Minie balls being undersized as they were, in a clean rifle, could and would shake out of the barrel if you carried it muzzle down. To avoid this of course drill stressed to never carry muzzle down. This was the whole advantage for the Cavalry with the development of breach loading weapons. Instead of carrying a muzzleloading musket with and undersize mine ball that would shake or slide out of the barrel while jostling and riding the breachloaders could load an oversize ball, tight fit that is, and powder charge behind it, close the block and have no worries of the loaded ball sliding out when your carbine was slung to your horse or you were carrying it one handed muzzle angled down.