In the X-Men movies whenever Rogue touched someone she would temporarily get their powers. And that is why she had to wear gloves all the time, but what I'm wondering is was it only her hands that had this "power"? Or could any part of her body come in contact with another mutant and gain their ability?
Rogue's mutant ability wasn't confined to her hands, but her entire skin.
According to the Marvel Universe Wiki
Rogue is a mutant who formerly possessed the ability to absorb the memories, abilities, personality, and outward physical characteristics of other beings through skin-to-skin contact. Such transfers lasted for 60 times longer than the contact time, with extended contact resulting in the possibility of permanent absorption. No upper limit had been determined for the number of beings Rogue could simultaneously imprint. Upon absorbing another's memories, Rogue also gained any associated emotional responses. Rogue was typically able to control such emotions, however absorbing psyches more powerful than her own resulted in Rogue’s psyche being supplanted.
In the X-Men movies, Rogue's power was to steal the life force from anyone she had skin to skin contact with. In the case of another mutant, she also stole their power. I believe the only one capable of surviving prolonged contact with her was Wolverine.