Don,
This unit was composed of 1 officer and 25 other ranks from each of the 13 infantry brigades of Washington's army. As such they would have had no flag and no common uniform (if any at all – pretty much everyone fought in their shirtsleeves on the day). They were not a "punishment" unit as suggested by Tom Bryant, nor should they be confused with the "select men" brigades of Wayne and Scott, although they were raised in a similar way.
The "single battalion of "picked men" was allegedly created from men selected on the basis of their marksmanship. This single unit was attached to Morgan's flank corps which was supposed to attack the vulnerable British wagon ,train from the south. Unfortunately, they missed their cue because Lee's orders told Morgan to attack "tomorrow" without realizing he had written the orders after midnight and should have said "today". Morgan's command comprised about 200 Monmouth county militia in three or four battalions, and around 600 Continentals – about half of them the "Picked Men" and the other half from the four VA and three PA rifle companies and two NC light companies that had been with him for some time.
At the end of the Monmouth Courthouse campaign, all of these men/companies returned to their parent battalions and any common membership of subsequent light infantry corps is coincidental (that said, some of the Advance Guard battalions of "picked men" were commanded by officers who had been part of Maxell's Light Corps in 1777). The original battalions of light infantry that fought under Maxwell in 1777 had all been disbanded and again, bore no relation to these 1778 units, or those formed in 1779 and afterwards.
Rating them as "elite" is problematic, because they never actually did anything during late June/early July 1778 (for different reasons, it is equally difficult to rate the "select battalions" of Wayne and Scott who formed part of the Advance Guard under La Fayette initially, and then Lee). In theory, the picked men were marksmen, but given the numbers of men detached from the regular Continental battalions throughout this campaign, it is hard to say how good they were. To put this inperspective, some of the rifle companies assigned to Morgan had to borrow rifles from other units and the central stores in order to equip themselves completely.