Pick up your chicken calmly, hands covering the wings and place its head between your left arm and your body facing to the rear. Now bring your left hand under to grasp its feet.
Now with your right hand reach over and spread its feathers away from its rear end. There, for your examination, is the vent.
You should examine this area occasionaly, as it is the business end of the hen and so, rather important. Check for pliability of the surrounding flesh and to see that that pubic bones are well spread - you should be able to get 2-3 fingers well pressed in between them.
And yes, both egg and poo come from the same orifice.
May I suggest that you obtain, at the earliest convenience, a good book on chicken rearing? There is no substitute for a decent reference, close to hand.