Why did people swallow cotton balls to avoid the draft?

Answer 1

Well, honey, back in the day, some folks thought swallowing cotton balls soaked in Orange Juice would make them fail the military physical by causing a fake medical condition. Spoiler alert: it didn't work. It was just a silly and ineffective attempt to Dodge the draft.

Answer 2

The belief was that swallowing cotton balls would show up as a shadow in x-rays during the physical examine. Thus eliminating them from service...as a shadow would indicate an underlying health problem, such as a tumor.