When the (Army) Doctor grabs your balls and asks you to cough, what is he testing for and what do your balls do if they are healthy/unhealthy ?
He’s not grabbing your testes, he’s poking your abdomen. He’s looking for signs of a hernia.
No, he definitely grabs your balls. Maybe we just encountered strange doctors. So what happens to your balls if you have a hernia?
They’re supposed to twitch upwards when you cough.
Perhaps the doctor was also tossing in a testicular cancer check along with the hernia check.
Science teacher (former) checking in…
When you were a wee lad, still developing in utero, your “fellas” were developing up inside you. (roughly the same position where the fairer sex have their ovaries) at some point before birth (hopefully) your “jewels” dropped into your scrotum. To do this, they had to pass “through” the muscle at the base of your pelvis. Luckily for you, this muscle has openings for this reason. This muscle of which I speak is the one you tighten when you do Kegels. (like you just did!) Yes, they work for guys, too.
At any rate, these openings are supposed to close to a much smaller size after the “grapes” descend. If they don’t, part of your kishes can get pushed through the hole later in life and can get pinched (hernia).
Coughing? Well, when you cough, you tighten you abs and diaphragm. This puts pressure on your guts and makes the herniated piece of intestine push out a bit. When checking for a hernia (and you can do this too) he/she 
“pokes” a finger into the top of the scrotum, where the muscle opens. If your doctor is gently cradling “the twins”, you may want to consult your “Preferred Provider” list again. (Unless she’s a hottie 
)
Se non e vero, e ben trovato.
Did I mention that I taught chemistry and physics?
When I had my last phsical, I asked my doctor exactly this question. He said he was checking for hernias.
I’ve done thousands of physicals and hernia checks, and I’ve never grabbed someone’s testicles and asked them to cough. I check for hernias by jamming a finger into the inguinal canal above each testicle and ask them to cough. OK, perhaps its not pleasant either, but it’s a world of difference away from the OP’s experience. I do palpate the testicles, epididimi, and other scrotal structures during the exam, but I know of no reason to hang on to them during a cough (no legitimate reason, that is!).
Qadgop, MD
This question reminds me of a draft board physical. There were at least 500 guys, naked, shivering and standing shoulder to shoulder in lines. A doctor commanded each line in turn to “Bend over and spread your cheeks!”. The view was lousy for everyone but those of us in the front row. The doctor then went down the line with two medics, one with a large box of gloves the other with a bucket. The man to my right knew they could do nothing to us, so he bent over and grasped either side of his mouth and “spread his cheeks.”