The flags carried by the regiment are incorrectly shown as rectangular. Both national and regimental flags carried by Union infantry in the Civil War were square.
The enlisted soldiers in the film are shown wearing the rectangular Model 1851 belt plate instead of the brass oval "US" belt plate worn by enlisted soldiers.
None of the soldiers is wearing the proper cartridge box which would have been slung over the shoulder and which had a round "Eagle" breastplate.
When the 304th launches its attack, a bugle plays the cavalry command for charge. During the Civil War Union infantry had a completely different bugle call.
In close-ups of Henry Fleming, particularly in the initial engagement with the enemy, the lock plate on his musket is apparently stamped near the rear with a British Crown. While British arms were used by the Confederates lacking sufficient arms production capabilities, Fleming's group would have been issued US-produced rife muskets. In a close-up of another soldier, the US eagle stamped on the lock plate is clearly visible.
At the end of the film, as the Union soldiers are marching off, one of them notes that the days are getting shorter and another one wonders if they can be home in time for spring planting. This makes no sense, since the narrator at the start of the film explicitly noted that the story takes place in Virginia in the spring of 1862, when the days would have been getting longer, and clearly too late for the soldiers, who are from Ohio, to be home in time for the spring planting.
All the soldiers in Audie's infantry outfit have crossed rifles on their forage hats. The crossed rifle insignia was not adopted by the US army until the year 1876, before this it was a hunter's horn.
Beginning at 14:28, several Union soldiers wading across the waist-deep river and later climbing out of the river are carrying the U.S. Rifle Model 1903, also known as the Springfield Model 1903. Note especially the soldier who climbs out of the river holding his rifle above his head. The Model 1903 is a bolt-action, magazine-fed rifle that was adopted by the US Army in 1903 and used in World Wars I and II, but not the American Civil War which was fought from 1861-1865.
The drilling and action center around one platoon. While the platoon did exist in Civil War regiments (as half of a company), in practice it was rarely used. Units quickly lost enough personnel due to illness (if not combat) that companies were the smallest practical unit.
When the unit first moves out, in the spring of 1862, the fife & drums play "When Johnny Comes Marching Home Again". The melody of that song is from July 1863, and the song was published in September, 1863.
The movie was shot in California and the brown grass and oak trees are obviously not the tidewater area around the Rappahannock River in Virginia where this story is set.
There was no 304th infantry regiment in the Civil War. No state assigned a regimental number over the mid-100s.