I sent Dave the info in a Private Message, for the reasons stated in it, below. Let me publically compliment him for taking so graciously the information which contradicted his prior belief about who wore his eagle-A button, and when. (Believe me, many people don't take the contradictory info very well, even when it is politely phrased.) He said I should post the contents of the Private Message, so here it is.
=================
Dave, I'm sending you a Private Message because I don't want to blindside you in public by giving you a "contradictory" historical analysis of your Fort Myers eagle-A button. I'm writing to you because as a civil war relics digger myself (for over 40 years) I prefer to research out the actual facts as far as possible, and "let the chips fall where they may." You may feel differently, and if so, I'll apologize and drop the subject.
The "form" of the eagle on your eagle-A button, and more importantly its "Scovills & Co. / Waterbury" backmark, date it to the 1840s. Scovill used the Scovills & Co. name only from 1840 to 1850, when it became the Scovill Manufacturing Company.
Being made in the 1840s makes it very-very unlikely that your button was issued to a soldier during the civil war... especially in the latter half of the civil war. All the pre-war stocks of eagle buttons got used up in the first year of the war.
Also, at the time of the civil war, having a letter in the shield meant the eagle button was for Officers, not Enlisted-men. If it was worn by a free black artilleryman during the civil war, he'd have to be an officer, in command of a section of the artillery battery. So far as is known, there were no black Artillery Officers. The only black soldiers at Fort Myers were one company of troops from the 2nd US Colored Infantry regiment. They served the Fort's cannons during the 1865 battle, but they were not wearing uniforms which had artillery buttons.
All of that makes it more likely that your button was issued in the 1840s (or because Scovill used the Scovills & Co. backmark into to 1850, perhaps it was issued in the early-1850s) to a soldier who was part of the garrison of Fort Myers at that time. We know that Fort Myers was built in the 1830s and was "used as a base of operations against the Seminoles" during the three Seminole Indian Wars (in the 1830s, 40s, and 50s).
I hope I have not offended you by providing you with the historical evidence about your button and the "timing" of its presence at Fort Myers. Like I said, I myself would rather know the actual historical facts about what I've dug, than think something that is incorrect about it.