Sorry this took so long for me to get to!

I think your question makes some assumptions about the nature of pre-modern combat that might not necessarily be true. It is easiest to think of battles as large blocks of men crashing into each other (or later, shooting at each other for a bit before crashing into each other). Many history books descriptions of battles from the period in question will also do the same. It makes for prettier maps, and puts the focus on the commanders who names were attached to the big blocks of men. However, this way of viewing battle does not necessarily align with the experiences of the fighting men.

Take, for example, the Roman legion. A legion was made up of thousands of men, fighting in disciplined ranks. The smallest organizational unit in the post-Marian army, however, was a contubernia of ten men. 6-10 of these made up a centuria, and six of those made a cohort. While these units fought as part of a larger line, it shows that the Roman military thought about their army in a more granular way, and that duties combat and otherwise could be assigned to squad-sized formations. Look into the writings of Adrian Goldsworthy for more on how the Roman armies fought.

The same is true during the American Revolution. In an ideal battle, two armies would engage on a flat, open field with no obstructions. In this kind of fight, the regiment would be major unit of engagement, with light infantry battalions fighting on the flanks. Of course, no battle ever actually occurred on so perfect a field. Many of the Revolution's most famous battles - White Plains and Guiford Courthouse spring immediately to mind - occurred in broken, wooded terrain. In With Zeal and with Bayonets Only, Matthew Spring recounts a story where the decisive British charge at White Plains began when a non-commissioned officer, realizing his platoon was pinned down and vulnerable, bodily threw his 16-year-old lieutenant over the stone wall they were hiding behind and told the surrounding men to "follow the young man's example." Long, Obstinate, and Bloody portrays Guilford Courthouse as a confused affair where regimental lines broke down and fighting became basically tree-to-tree.

Of course, it's also wrong to say that the modern military fights exclusively in squads. A modern squad has more firepower, autonomy, communications, and mobility than in any other point in history, but actions are nearly always coordinated with bigger organizational bodies and other friendly assets in an area.

I'll ping /u/tiako and /u/georgy_k_zhukov, who can add some depth to these topics.