A stress ball is a malleable foam ball, usually squeezed in the hand,  to help relieve stress.

It was important to make people aware of this product and its qualities as well as create a stress free campaign. Stress, by research, has started affecting people at a very young age and hence, this campaign was aimed at a younger age group. Keeping this in mind, the brand name, ‘BALLS’ was chosen, it is a slang, much used and hence was used to capture attention.
Stressors that were quite odd and do not trouble most people as well as creatures and scenarios that the target group know of and some even stress about, for example the zombie apocalypse were employed to advertise. 
The copy in each ad puts into perspective the meaning of each situation.
Yes, the pun in each advertisement is very much intended. The advertisements with creatures like -
ex. The Vampire, which are not known to exist, show the vampire saying Balls to the ‘Cross’, hence, defying an age old myth about the fact that vampires dread the presence of crosses and other holy objects for it causes hurt and even death by burning.
The copy says burnout. Burnouts are mostly experienced due to job stress but may also be caused due to other stressors. Hence, they are interlinked. 

So, use Stress[balls].
It’ll help you deal with anything - 2012, alien invasions, zombies, vampires, asteroids, and yes - even that annoying girlfriend who doesn’t shut her mouth and goes on and on and on and on and on and on.
Now stop thinking about her.
Say BALLS.