Have some slack in the sheet and don't be afraid to throw the egg hard, but don't squeeze it. A fitted sheet makes catching the egg easier. A water balloon can be substituted for the egg if you're nervous. It is very difficult to twist a spinning bicycle wheel, especially if the tire of the wheel is filled with sand so the wheel has a large angular momentum that needs a large torque to quickly change its direction. The wheel needs handles fastened to its axle on either side.
Commercial versions are available.
Be sure to wear a glove on the hand spinning the wheel and pick someone with fairly long arms and no loose clothing. Bike Wheel and Lazy Susan Hold a spinning wheel while standing on a turntable. Skater on Lazy Susan Hold a sandbag in each hand out at arms length while spinning slowly on a turntable. When you bring your arms into your chest your rate of rotation increases showing conservation of angular momentum. Have your audience just consider the speed of the sandbags.