That is a remarkably... specific question...

The Greek sources after Homer tell us very little about specific ways to incapacitate people in combat, and most of our modern assumptions are based on reconstructions of how the equipment they used would protect someone and how a warrior might have the best chance of inflicting a wound.

Since Greek heavy infantry went into battle carrying a large round shield in front of him, it has long been assumed that the only effective way to hurt an opponent was to go for the thighs (below the shield) or the throat (above the shield). However, Christopher Matthew has recently done a lot of work comparing combat depictions, actual remains of ancient armour, and "experimental archaeology" (testing out replicas), and his conclusion is that the notion of a throat "kill shot" is probably a myth. Excavated armour that shows combat damage has usually been struck in an upward direction, suggesting straightforward underarm thrusting rather than overhand "poking" over the rim of the enemy's shield. The most common targets for spear thrusts of this kind were the enemy's centre mass - his chest, straight through the shield - and the head that poked out over the shield.

However this may be, it is a fact that the meat and two veg were essentially unprotected by even the heaviest Greek hoplite panoply. Only a few strips of hardened linen served to deflect blows to the hoplite's other spear. A warrior could do no more than use his shield and his distance from the enemy to keep his manhood safe. Therefore, whether by overhand or underarm thrusting, blows aimed at a man's junk may well have been common; it was a "soft target" in more ways than one.

Amazingly, we actually have an ancient reference to a man struck in the nethers and bleeding to death from the resulting wound. The 7th century BC Spartan poet Tyrtaios wrote these immortal lines:

An aging warrior cut down in the vanguard of battle

disgraces the young. His head

is white, his beard is grey, and now he is spilling

his powerful spirit in the dust,

naked, clutching his bloody groin; a sight

for shame and anger. But youthful

warriors always look good, until the blossom

withers. Men gape

at them in life, and women sigh, and dying

in combat, they are handsome still.

Some scholars have taken the line about the man's wedding tackle to be a reference to mutilation of the war-dead; his privates must have been cut off for him to be able to hold them in his hand as he lies dying. But this is of course unnecessary - most people can reach their family jewels just fine even when they are attached. More likely, then, the man was stabbed in the twig and berries, and the wound was lethal. He serves Tyrtaios as an example of a graceless death; old men do not make great warriors, and the young should strive to do the work for them, knowing that they will have beauty and glory even if they fall.

Edit: also, yes, the Greeks were fully aware of the phallic nature of spear-fighting. One of Ares' epithets was "the Penetrator".

Edit 2: this post about getting shanked in the balls is now my second most upvoted post ever. Thanks a lot, Reddit

There is a rather odd law in the Old Testament which deals with this (Deut 25:11-12):

11 If two men get into a hand-to-hand fight, and the wife of one of them gets involved to help her husband against his attacker, and she reaches out her hand and grabs his genitals, 12 then you must cut off her hand – do not pity her.

I'm not aware of any corresponding law applying to men. Without context it is not clear whether this was seen as an unfair tactic in general; if it was about a second person entering the fight; or if it was strictly to do with gender roles.

The date and place of composition of Deuteronomy is disputed, but a common view that the core was written in Jerusalem in the late 7C BC, some composition during the Babylonian Exile, and the final redaction about a century later after the return from the Exile. A more traditional view would put the core composition earlier.

The ancient Greeks' most violent sport, pankration, was as close to no-holds-barred combat as you could get outside of warfare. It was contested in the Olympic Games for about 1000 years, so there are many accounts about rules, techniques, and foul play. The players fought naked so the groin could certainly be targeted. It was not uncommon for players to be killed in competition, so it was certainly an extremely vicious event. The rules only explicitly disallowed biting and eye-gouging, so groin shots were legal. Kicks and punches to the groin were definitely part of the repertoire, but don't otherwise get a huge amount of attention, so the techniques may simply have not been as effective against a trained fighter as they are against a street fight opponent who is not expecting it. The stances of trained grapplers tend to make the groin one of the harder areas to target, after all. But in the end, pankration is not warfare, so all it really tells is is that groin shots were not seen as particularly taboo.

Somewhat related, but was something like a jock strap or protective cup a common piece of equipment for professional armies?

Follow-up question: If memory serves, aren't there pieces of artwork from these times illustrating battle scenes clearly showing soldiers giving their enemies a good kick to the groin? I believe I can recall seeing manuscripts illustrating fighting techniques employed during medieval times and this was one move that was used to incapacitate their opponent.