On June 1, 1995, about 1200 eastern daylight time, an American AA-1, N6241L, collided with the terrain, during the landing roll, at the Country Squire Airport, near Rock Hill, South Carolina. The airplane was operated by the pilot under the provisions of 14 CFR Part 91, and visual flight rules. Visual meteorological conditions prevailed. No flight plan was filed for the personal flight. There were no injuries to the commercial pilot, and the airplane was substantially damaged. Origination of the flight was Huntersville, North Carolina at 1120 eastern daylight time. According to the pilot, he entered a left pattern for runway 09 at Country Squire Airport. He observed that the wind was out of the south at approximately 10 knots. He stated that he made a full flap landing on the first quarter of the wet, sod runway. He held the aircraft on the center of the runway with rudder and brakes while holding right aileron into the wind. The pilot indicated that the roll out was normal until reaching the last quarter of the 2600 foot runway. The aircraft suddenly veered right and collided with terrain. The aircraft came to a stop ten feet off the runway. An airworthiness inspector from the South Carolina Flight Standards District Office inspected the aircraft after the accident. Both brakes operated as intended. He noted that the airplane is equipped with a castering nose wheel that is non-steerable.