Your instinct that they were possibly Southerners is correct, as both Mays men served in Company K of the 40th Mississippi Infantry Regiment, Abner as a private and Jesse as a corporal. Both were killed on the 19th of September 1862, according to their records.
There is a project that I'm working with Kenneth Byrd (originator of this thread), to identify as many, as possible of those Confederates buried in the mass grave at Shady Grove Cemetery, Iuka, Miss., in the hopes that a memorial of some sort may be created. I'm personally grateful to you for the additional information on the Mayses and the citation of its source. We'll add that to the data we've been collecting, and am also extremely grateful to have a local Iuka resident interested in the project!
Sources indicate that there are approximately 265 unidentified Confederates buried in that particular "trench" at Shady Grove. To date, we've positively confirmed the identities of 85 KIA, and another 17 men who died of wounds or disease at Iuka within a matter of only 5 days after the battle, for a total of 102 men who'd be classified as probably/most likely interred there.
Additionally, there are two more unconfirmed KIA (can not locate any surviving service records for these men) who were named in period newspapers as having been killed in that battle (on lists with confirmed KIA, wounded, missing, etc).
We've found and confirmed the names of another 30 men who also died at Iuka during September-October 1862, who MAY be buried there. Plus 10 other names whom are known to have died while at Iuka, but for whom we do not know the dates of death.
Still working on a few more leads, but we're closing in on as definitive a list as we possibly can. When it's "finished," we'll post it online, and furnish it to Iuka, and share it with the National Parks Service Interpretive Center at Corinth, as well.
-Chip Culpepper