Okay, let's make a list and start whittling it down, of American Unique Units, or at least American Units That Have Unique Properties Of Some Kind.
Roughly chronological:
Ground Combat Units
Rangers. - Roger's Rangers predates the USA, being a 'colonial' unit raised during the 7 Year's War (French & Indian War in the British colonies in NA). Would be, then, an early Industrial Era Recon unit, but of course might require a new title for the 'regular' Ranger in the game.
Riflemen - Morgan's Riflemen for a specific unit, although the 'over-mountain militia' of east Tennessee and western Carolinas who massacred Ferguson's British force at King's Mountain are another example: a Revolutionary War-era Recon Unit that reduces the combat factor of opposing units (by shooting down their commanders faster than they can give orders) - Berdan's Rifles would be the US Civil War version of this same unit.
Anthony's Legion - Anthony Wayne's reorganization of the entire US Army in the 1790s into a set of brigade-sized combined arms forces, each with regular infantry, riflemen, and light cavalry. An Industrial Era unit that could combine bonuses of Recon and Melee units.
Mounted Rifles - All the US cavalry of the 19th century (Industrial Era) were much more flexible than European Armies' - more willing to dismount and fight, able to charge, scout, pursue, or defend, and that was from their first start in the 1830s to the end of the century and beyond. "Mounted Rifles" was the most unique official designation (applied to 2 of the first 6 mounted units formed) but more generally, the 'Rough Rider' could be replaced by the Yellowleg, the nickname for all the US Cavalry. A Light cavalry unit that is slightly faster, no one gets any Anti-Cav bonus against them because they can fight you dismounted as well as mounted, and they have extra sight range like a Recon Unit.
Armored Cavalry - Modern Era Recon Unit. The WWII US Army Armored Cavalry Groups were regimental-sized units of light tanks, armored cars, jeep-mounted scouts, self-propelled artillery and antitank guns that could scout, pursue, defend: huge firepower of automatic weapons, faster than almost any other contemporary ground unit. Also were used a lot to secure routes of advance, enforce traffic control and so could provide extra movement to any unit that starts adjacent to one.
FDC Artillery - permit an ex-artilleryman this one: in the late 1930s (late Modern - beginning of the Atomic Era) the US Army developed the Fire Direction Center to control artillery fire: the result in WWII and later was that US artillery could mass and shift fire faster than any other guns in the world. Effectively, one artillery unit could support several infantry or armor units and change who they supported almost at will. One German Tactical Note in 1944 put it succinctly: any attack on an American Unit had to succeed in the first 60 minutes or it was Doomed: by that time every American artillery piece within range would be blasting the attack to bits. Any US Artillery Unit can fire twice in a turn if they don't move (this is probably 'way OP, but it would be accurate)
Sherman Tank - IF we must, the M4 Sherman's advantages were its incredible flexibility: it was built with welded or cast armor, motorized with gasoline, diesel, or converted aircraft rotary engines, used volute spring, modified spring, or torsion bar suspension, mounted everything from a medium 75mm to a long barreled 76mm to a British 17-lber (76mm) to a light 90mm cannon or 105mm howitzer, and, compared to any other medium tank of its era, was much more reliable and therefore likely to actually get into battle without breaking down (average serviceability rate for ALL German tanks in WWII was 60 - 66%: they had to build 3 tanks to get 2 of them into action!). Unique abilities then, would be recover damage 10 - 20% faster and recover every turn regardless of other actions, take fewer resources and gold to build and maintain.
Air Cavalry - Atomic Era Helicopter Unit - the US Army was the first to form brigade and division-sized units built around the helicopter for transport, fire support, resupply, and mobility over all types of terrain. Whatever other quality they have, they also have very high Maintenance costs, reflecting a huge maintenance 'tail' required to keep them operating.
Special Forces - the 'Green Berets' of the 1960s and later are iconic, but they were based on Britain's WWII-era Special Air Service (SAS) or Special Boat Service (SBS) units, so strictly speaking are not peculiar to the US military.
Not-Exactly Ground Combat Units: Semi-Civilian Units.
Homesteader - a Settler that can defend itself and is cheap to produce
Naval Combat Units
44-Gun Frigate - the USS Constitution and her sisters, able to take on 6th-rate Ships of the Line (Constitution shot the 50-gun HMS Java to bits)
Steam/Armored Frigate - the USS Kearsarge and her sister ships of the US Civil War were really unique: sailing frigate designs that had been 'upgraded' by mounting heavier rifled explosive-shell-firing cannon, steam engines, and some hull armor: an Industrial Era Upgrade to your 'regular' Frigates.
Fleet Carrier - the US CVA-type "fleet carriers" of WWII could carry and launch faster more aircraft per tonnage than either their British or Japanese counterparts, had better damage control and, after the middle of the war (1943), far more capable carrier aircraft than anybody. Perhaps an Aircraft Carrier replacement with better air and melee defense and a plus factor to any aircraft unit carried.
Nuclear Carrier - the modern CVN "Supercarriers" are unique to the US Navy, but also are late Atomic Era or Information Era units, and so of marginal usefulness in most games.
Air Combat Units
Civ games have already had the B-17 Bomber and P-51 Fighter and, indirectly, the B-29 bomber as the graphic for an 'atomic bomb' delivery. That doesn't leave a lot of iconic and unique aircraft types: later ones like the A-10, B-1, B-2, F-15, F-14 or F-111 are all going to run into the same problem as the CVN: available too late in the to be very useful in most games.
Lots more could be added, but that's a start, I think.