Additionally (if this is allowed), the full quote also bans "deserters from the Ministerial Army". What was the Ministerial Army, and why were they apparently known for deserters?
The Oxford English Dictionary supplies references in the use of the word 1681-1855 to mean someone with no fixed abode, a tramp or vagrant. ( like John Vanburgh's 1694 The Relapse :"I'm a Justice of the Peace, and know how to deal with Strollers") Racism aside, the reason Washington would not want any "stroller, negro or vagabond" in the Army is obvious: they'd uncertain loyalty and commitment to drill and fight, as the main reason they'd be in the Army is to be provided with someplace to sleep and a meal.
However, there is a second 18th c. meaning: "strolling players" were a troupe of traveling actors. Aston was an actor, and that title would have carried a nice bit of double meaning, if that was the title. But I don't know if it was. That just looks like the title of a book about Aston written in 1920 by Watson Nicholson, and I haven't read it.
Related question.
Did Anthony Aston write an autobiography titled "Anthony Aston, Stroller and Adventurer."
Or was his biography titled that in the modern era?
The history section of the U.S. Army website uses "ministerial army" to refer to the British Army, i.e., the one sent by the ministers in parliament.