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:
1. 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!
2. 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.
3. 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"):
1. ENVG-III, with thermal imaging which can see through smoke. This one piece of gear can be a total game-changer.
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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.
3. M203 grenade launcher. This turns every M4 rifleman into a grenadier.
4. 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. 347