Background
Before we answer the question of: "How best to use modern troops?" we need to understand the fighting conditions of the day. One of the most important facts is the engagement distance. Although both sides had rifles with a theoretical effective range of 400+ yards, most engagements happened at 50-100 yards! There are several reasons for this fact:
- Gunpowder was a severely limiting resource. Many soldiers were not proficient marksmen because neither army could produce gunpowder at a sufficient rate to allow them extensive shooting practice. Some soldiers were observed shooting their rifles at a 45 degree angle into the air, for lack of actual marksmanship experience and understanding of ballistics!
- Black powder turned the battlefield into a giant smoke cloud. Smokeless powder was one of the most important innovations in warfare, because it allowed soldiers to see the enemy even after many rounds of rifle fire. Soldiers often closed to 100 yards or less simply because they could not see what they were shooting at in 400 yard rifle range.
- Optics were primitive. Iron sights were generally the only optics available, and a lack of practice shooting means that most soldiers did not know how to properly account for bullet drop over longer ranges. In the 50 yard range for typical engagements, bullet drop was not an important consideration.
Now, cannon were available, and although they had an effective range of a mile or more, they were typically used at 1000 yard ranges or less, due to accuracy and other battlefield conditions. But you have to realize that artillery pieces were massive resource hogs, often being manned by teams of 10 soldiers, multiple horses, and special wagons called "caissons" which carried the ammunition and powder.
In addition, both sides sometimes employed war balloons to provide forward observation for artillery targeting. Obviously, this provided a significant advantage to look behind the smokescreen created by one's own rifle line, as well as identify targets of opportunity behind the main line.
Orders were typically given verbally to infantry units, and by battle courier to officers and NCOs.
Equipment
You said that the modern soldiers just have "basic" gear. Given the facts above, I would strongly suggest that you equip them with the following gear, which is more or less "standard", depending on the unit (but none of it is "exotic"):
- ENVG-III, with thermal imaging which can see through smoke. This one piece of gear can be a total game-changer.
- ACOG sights, which are designed for effective targeting out to 800 m. No Confederate officer will expect a Union regiment to fire on his troops at 600 yards. At least not with any effect. Your riflemen with their M4s + ACOG will effectively all be snipers under Civil War battle conditions.
- M203 grenade launcher. This turns every M4 rifleman into a grenadier.
- M18 Claymore mines. Optional, but a massive force multiplier if used well.
Defense
One of the most effective use of the modern soldiers is defensively during a large pitched battle. You'll want all of the troops to be equipped with the thermal imaging goggles for battlefield visibility. That's expensive, but you only have 300 of them, so no problem, right? I would then assign targets as follows:
- Snipers: target balloons, then officers/color teams, then artillery
- SAW gunner: start fires at 6-700 m starting at the flanks and working towards the center until enemy is defeated
- M4 riflemen: engage single targets with precision fires (single round at a time) at 500 m, then switch to M203 at 350 m if there are still massed troops approaching. Otherwise, continue with precision fires.
Before the battle, you want to prepare the field with claymores, if you have them. They should be placed in the middle of each formation where enemy troops are expected to charge, about 150 m from the front lines (i.e., before the typical engagement distance). It's more effective to place them densely in the center rather than spreading them out, for reasons which will soon be clear.
When the fighting starts, the SAWs will start picking off troops at the flanks of each formation. The enemy will quickly learn that that is a dangerous place to be, and will charge towards the center for "safety". Union soldiers can make a show of faking casualties towards the center, or even fleeing it for the flanks, to make it look like the Confederates will be able to break the lines if they push forward. Warfighting theory at this time was heavily predicated on breaking through enemy formations to get behind the lines and wreak havoc, so this will be a natural instinct for many soldiers.
Of course, what we are really doing is creating a kill box for the M203s and the claymores. The SAWs should be able to thin a pretty good number of troops in the 500-800 m range. They are modern weapons designed for the modern battlefield where suppressing enemy troops hiding in cover is the task of the day. Against unarmored troops charging in the open, they should simply annihilate any charge almost by themselves. But let's say there aren't enough to finish the job by themselves. The M4s should be able to lay down effective and deadly fires in the 350-500 m range, thinning the charge dramatically. At at 66% accuracy, the riflemen should be able to hit about 20 soldiers per magazine, or about 140 per 7-mag loadout. The SAWs at 50% accuracy should be able to pick off about 100 soldiers per 200-round belt of ammo, or 400 soldiers for a typical loadout.
If we assemble 10 squads of 8 riflemen + 1 SAW + 1 sniper, then each squad should be able to average maybe 1500 kills each. Now, that's only counting rifle kills. If we consider each M4 rifleman carries 4 grenades with a 130 m casualty radius, and assume soldiers are bunched up at 2 m apart, then each grenade may be able to cause 60+ casualties, or 240 per loadout. That's another 1920 casualties per squad. If we add 2 mines for every soldier, and we assume each mine can injure another 25 enemies, then that's another 400 casualties per squad. That's nearly 4000 casualties inflicted per 10-man squad!
The average Confederate brigade was a force of about 5000 soldiers, so given that some soldiers will inevitably break and run, we can safely surmise that a single well-armed squad could defeat an entire Confederate brigade all by itself. If you add on Union soldiers picking off the stragglers, it seems likely that a squad + Union brigade could easily take on 2-3 Confederate brigades.
I say the Union soldiers would become far more effective because the modern soldiers would be firing on the enemy and inflicting casualties long before the enemy got into typical engagement range. So the Union soldiers could mostly hold their fire until enemy troops actually closed to the 50-100 m range, giving them a clear view on their side. The Union also had superior artillery support, and could unleash devastating volleys at the remaining attackers.
A better use of Union soldiers would be to push them forward in a skirmisher line. They could engage from a prone position all the way out to the line of claymores (which obviously they want to stay behind), making their effective range about 200 m. They should stay mostly on the flanks to funnel attackers into the killzone, and keep their heads down so the modern fighters can shoot past them (although, the Union often had the high ground, which makes shooting over them much easier).
Offense
After some devastating defeats using the tactics above, the modern soldiers should strike out on their own as a few special hunter-kill companies (up to 5). Catching a Confederate division on the march with the cover of forest would very likely cause them to scatter and flee, leaving behind their artillery and most of their supplies. Since the Confederates were already struggling with supplies, this kind of harassment would likely deplete the warfighting capability much faster than open combat. There is no evidence that Civil War era soldiers had any experience in insurgent-style ambush tactics or warfare, and commanders would struggle to organize their troops under such conditions. Again, claymores and snipers can be used to great effect by preparing the battlefield and targeting officers, color guards, even horses. Artillery can be rendered completely useless by lobbing a few M203 rounds on the caissons.
But hey, why expose yourself to enemy fire by attacking in the day? We've already equipped the entire force with NVG, so launch the raid at night, starting with that artillery powder, to make some nice bright startling explosions to wake everyone up at 2 AM. Even modern militaries know that the US owns the night, and that is when they are most vulnerable to attack. A Civil War era general will literally not know what hit him and will struggle to mount any kind of effective defense. It would pretty much be shooting fish in a barrel. At that point, let the M249s loose on rapid fire and don't worry too much about conserving ammo. Most soldiers would probably flee in the night hoping to escape with their lives. Let the general fight the demons and ghosts who can see in the dark!