Scott,
I did not find a listing for James H. Shegog in the Mexican War Pension books I have; nor for a widow or minor children.
I suggest you ask the National Archives for his Mexican War compiled service file and his Mexican War Bounty Land Warrant Application filed for service in Co. E, 1st TN Inf. If they do not find those records, then he probably didn't serve in the Mexican War in that company.
If you do confirm he was a Mexican War soldier, the Descendants of Mexican War Veterans (
www.dmwv.org) would like to know where he is buried and if there is a stone marker.
Sandy