A cylinder would allow all geodesics, but then it would still have two 90 degree angles and two 270 degree angles so still not a square. I think it would be a trapezoid/trapezium, and might be a parallelogram depending on what definition you use.
There might be some crazy custom shape that makes the angles on the more complete circle segment actually 90 degrees but I don’t think there’s a common easy-to-conceive shape that works.
A cylinder would allow all geodesics, but then it would still have two 90 degree angles and two 270 degree angles so still not a square. I think it would be a trapezoid/trapezium, and might be a parallelogram depending on what definition you use.
There might be some crazy custom shape that makes the angles on the more complete circle segment actually 90 degrees but I don’t think there’s a common easy-to-conceive shape that works.