Eating the cucumbers with the skin intact is actually healthier than eating the cucumber without the skin.
The skin of the cucumbers can also help protect against colon cancer and the skin of cucumbers also contain fiber which can help prevent constipation.
Cucumbers are also a good source of Vitamins Potassium and Antioxidants.
I always leave the skin on the Cucumbers I eat because I love the skin and the cucumbers together and they're healthier that way with the skin intact.