Haha, it is obviously very informal. Do you have more context?

It's just a funny way of saying that someone has supported you and protected you. The usual saying is "covering my back", where "cover" means protect. Balls - that's slang for a man's genitals (testicles) which can be injured very easily. So "covering my back and balls" just means watching out and protecting you from anything that might hurt you.

I've never heard of this expression, and didn't find any results on Google. It sounds vulgar, but the context you provided sounds formal, so I'm confused. If I had to guess, I'd say it's an emphatic way to say "cover sb", which means help/replace sb when they're unable to carry out a task.

I especially want to thank Collene Carson, president of GRC, for covering my back and balls for the last twelve years.