They did indeed! Military strategy as derived from Napoleonic techniques influenced leadership in both the north and the south, and played a large hand in informing both army organization, and battlefield theory (for artillery, infantry, and cavalry). George McClellan would eventually lead the Army of the Potomac during the ACW, but before the conflict had started, he had already written and published two influential military works (i.e., manuals) on bayonet tactics, as well as cavalry techniques. The former was directly influenced by French military theory that he had personally translated, and the latter was the result of his time in Europe during the Crimean War as an observer. Both of these works were influenced by European theory (one of them specifically French), and had a huge hand in informing military techniques during the ACW.

Another highly influential author of military theory leading up to the ACW was Henry Halleck, who would also go on to command Union forces at the highest levels during the ACW. In the mid-1840s, Halleck was sent to Europe to absorb and learn new or evolving techniques, and he came back with lots to say. His 'Elements of Military Arts and Science' bore out a lot of Napoleonic influence, and informed many of the military commanders, north and south, on "proper" military techniques during the ACW.

In sum, the prevailing military theory of the 1860s was heavily influenced by European theory of the 1830s, 40s, and 50s, which was itself a student of the Napoleonic school. Although Halleck and McClellan were both outed as poor field commanders during the war, they nevertheless had an unmistakable influence on the persecution of the war during its early years. This influence was directly tied to European military theory of the early to mid 19th century, which was the product of Napoleon's influence in many ways.

[Sources: Bruce Catton, 'Mr. Lincoln's Army'; Shelby Foote, 'The Civil War: A Narrative, v. 1'; James McPherson, 'Battle Cry of Freedom']

They did, though not initially. At Bull Run (I like to call it Manassas, even though I'm from New York), the Army of Northeastern Virginia (this was the Union army, which would eventually be renamed the Army of the Potomac) faced off against the Confederate Army of the Potomac (confusing, I know... this was soon reorganized into the Army of Northern Virginia) and the Army of the Shenandoah. The highest level of organization in these armies was that of brigade, composed of several regiments apiece and numbering somewhere around two to four-thousand men. After First Manassas and the quagmire it had become, General George B. McClellan assumed command of the now-renamed Army of the Potomac. He was known as "Young Napoleon" by both his supporters and detractors and, true to form, sought to fill that role. He reorganized the Army into several Corps, which were then divided into divisions, brigades, and regiments. Thus, he sought to maximize the Army's ability to cover vast distances and still be able to give battle if enemy forces were encountered piecemeal.

The Confederates did not adopt this system until after the Seven Days Battles, which was the first battle (series of battles) where Robert E. Lee commanded the Army of Northern Virginia. He implemented the Corps system sometime after the Second Battle of Manassas in August 1862. From 1862 on, both sides used this system of organization in their armies until the end of the war. It proved to greatly increase battlefield and campaign mobility, as well as streamlined the command structure and improved the combat readiness and effectiveness of the armies.