A version of "Shule Aroon" (you choose your spelling) from 1824 mentions a "cap of steel." I see it has grown into a coat.

Bruce Olson, thread 7985: Shule said (1999) "I'll believe that bit about the wild geese origin for "Shule Aroon" and the Revolutionary War "Johnny Has Gone for a Soldier" when I see some real evidence for it."
We have had reams more of speculation and mis-information since then, but nothing to prove that these songs are anywhere near as old as some people speculate they are.

Joe Hickerson, formerly head of the Folklore Archive of the Library of Congress, told Bruce Olson that he knew of no 18th c. broadsides (of Shule A.) at the Library of Congress, and also was skeptical that it was sung in the US prior to the US Civil War. Thread 7985, linked above.