While technically just about everyone over a certain age can claim to be a veteran of the Yogzarthu War, capitalised it refers to the genetically-enhanced soldiers who were designed to fight the Yogzarthu hand-to-hand if need be.
The supersoldier program was funded by the Tharin military and run by three Aquaans: a pair of bioengineers named Rejene Ri'el and Ama Zaha, and Thalassa Partasah who served as a psychological consultant.
Veterans were never intended to be whole platoons of enhanced soldiers, but weapons designed to kill Yogzarthu with a minimum of collateral damage.
Veterans can be identified by their facial tattoos - it is illegal for non-enhanced humans to wear such tattoos, and illegal for the Veterans to have them removed unless they give up their enhancements. This was both so they would be easy for their commanders to identify, and also because it was thought that humankind in general would prefer they were easily identified. Most humans are perfectly fine with Veterans and are more likely to buy them a drink than to find them scary for their abilities. The sibyls draw a bit of prejudice from people who forget they're not mind-readers.
While it's theoretically possible that any person could have any enhancement, the war gave a need for efficiency and volunteers were sorted by genetic potential. This meant that biologically-related humans would almost invariably end up with the same enhancements.
The genetic enhancements are coded very specifically to each Veteran. This was so that the Yogzarthu couldn't easily copy it - one couldn't give itself super speed in any form, it would have to take on the complete DNA profile of the specific Veteran it copied from, gaining various human weaknesses in the process. As well, the enhancements cannot be transferred from parent to child - a child of Veterans will be an ordinary human.
The process is reversable. Many Veterans chose to become non-powered humans again after the war. Others kept their enhancements, whether out of fear that the Yogzarthu would return or just because they'd grown to like their new abilities.
Run by Aquaans, the program had certain Aquaan quirks. The primary one was the psychological testing, with the result that most Veterans are even-tempered, pragmatic pacifists who only use violence as a last resort. All impulsive, tempermental types, all the ones who showed pleasure at the idea of becoming living weapons, were removed from the program. Pacifists themselves, Aquaans don't trust weapons in the hands of people who want to use them.
After the war, Ri'el and Zaha shut down the program and hid their research. While they continued to work for the Tharin government, they refused to turn any more people into weapons.
Red stripe: Strength enhancement. The formula seems to be that a Strength Veteran is three times stronger than their natural DNA profile would allow. They have have denser bones and muscles than non-enhanced humans.
Blue stripe: Speed enhancement. Speed Veterans are able to move at blurringly fast speeds for a few seconds at a time. They are somewhat stronger than average, needing a certain amount of muscle power for their speed, but also because, come on, they're soldiers. Speed Veterans are slightly more common than strength.
Purple stripe: Psychics, known as 'sibyls' just to keep up the 's' theme. The sibyls were created as Yogzarthu detectors; all have a type of low-level aura sense so they can tell species instantly and accurately. The most powerful sibyls are, at most, empathic. Few have a range of greater than five kilometres. There were never many of them and less than forty are still active today.
All Veterans have a regenerative factor - a broken arm heals in minutes, for example. It also makes them immune to many poisons, almost all regular illnesses, and nigh-impossible to get drunk. As far as super powers go, Veterans aren't spectacular. They were given enough power to combat the Yogzarthu and no more.