Strictly speaking, the Roman legions did make use of a squad-like unit called a contubernium comprised of 7 soldiers, an officer called the Decanus, and 2 auxiliaries or slaves in charge of logistical support for the unit. However, the contubernium was primarily a logistic tool rather than a factor in battlefield tactics. The word literally means "tents-together," and designated a group of 10 men who traveled, camped, and carried their equipment together. On the battlefield, the 8 military personnel were grouped with 9 other contubernia to form a centuria, which was the base-unit of early imperial military activity.

For most ancient armies, there simply wasn't a practical use for such a small unit in combat. Ancient warfare was formation based combat. The strength of the army came from fielding a large and cohesive mass that could maneuver into place and strike with swords or spears all at once. Having many rows of soldiers ensured that the line was difficult for the enemy to penetrate, preventing the opposition from getting behind Roman (or Greek, Persian, etc) shields, to the relatively unprotected sides and backs of their infantry. A contubernium of 8 was just too vulnerable to being dispersed or surrounded and insufficient to assault any larger formation.

Compare that to the typical modern military structure, where the squad can be even further divided into two teams. One team takes the role of a largely stationary fire base, providing cover fire, while the other is a maneuverable element that can get to a new and more advantageous position. Modern armies also typically have a step between squad and company (the rough equivalent of a Roman centuria): the Platoon. Platoons typically include specialized weapons units for heavy machine guns or personal artillery. None of those subordinate functions were relevant to ancient warfare.

Lacking the accuracy or range of modern firearms, arrows and slings were only effective on the battlefield when they could be used en masse, necessitating larger units in ancient armies that employed missile fire as a core part of their doctrine. Outside of that specific context, there was no use for "cover fire" when an ancient unit wanted to maneuver into place because the enemy couldn't reach them from more than a few feet away at most. Likewise, there was no ancient equivalent to the heavy machine gun or the rocket launcher as a weapon that enabled single soldiers or small groups to deal profound damage on their own.

The one example of an ancient army employing similar battle doctrine to modern forces that comes to mind is more of an exception that proves the rule. In Xenophon's Anabasis, he describes how he, along with 10,000 other Greek mercenaries, fled through northern Mesopotamia following the defeat of their employer, Cyrus the Younger. They were pursued by a large Persian army, which repeatedly sent cavalry raids against the Greek line.

The 10,000 were in a relatively unique position in ancient warfare. They were never intended to face a fully equipped Persian army on their own, as they had simply been the heavy infantry component of a larger force prior to the Battle of Cunaxa. As a result, they were outnumbered and unable to combat Persian cavalry or missile troops in a pitched battle. Their flight through Assyria began when their highest-ranking officers were massacred, leaving them with a fragile council of lower officers sharing command. This council agreed that it was imperative that they keep moving no matter what, rather than get caught in a siege of some adhoc encampment deep in enemy territory. However, that also meant that they had to keep their backs to the enemy, the exact opposite of an ideal ancient engagement. They also had to maintain a tight formation with long columns and short rows. That kept their exposed rear to a minimum number of soldiers and ensured that everyone would follow the road.

The problem was that this long Greek column could be easily flanked, picked off, and disrupted by Persian cavalry armed with bows and javelins. Each disruption forced more men out of formation and into individual vulnerability to the same threat. To combat this, Xenophon and the other commanders instituted two novel, but temporary, reforms in their army. Their missile troops, particularly Rhodian slingers known for their long range accuracy, were mixed into the melee infantry lines to ward off Persians as they approached, individual sections of the Greek line. It wasn't quite cover fire in the modern sense, but served a similar purpose.

They also formed six companies of 100 men, each with their own commander, and gave them autonomy to maneuver out of formation as needed, allowing them to step off the road and turn to face oncoming attacks while the rest of the army could just march on, knowing that the company would handle the attack. Within each company, several squads of 10 were also appointed their own officers and given further freedom of movement to break off on their own. That included defensive actions, but also logistical tasks like clearing the dead from the road or falling back to help shift the overall formation when the road narrowed.

These tactics were invaluable in the unique scenario faced by the 10,000, where a poorly prepared army found itself in a perpetual fighting retreat against missile attacks and cavalry raids, but would have had little value in a true battlefield scenario. That exact dichotomy can even be seen in the later exploits of the survivors from the 10,000 when they reached the Pontic Coast of Anatolia, fought in Thrace, and joined the Spartan invasion of Ionia over the next few years. Once they were fighting pitched battles rather than defending their very long retreat, the same army reverted to a more typical structure.