Unlikely in a country like the US where the "citizen soldier" model is prevalent and veterans affairs are a constant sore issue.
We actively PREVENT soldiers from using performance enhancing drugs in most cases (we actively test for anabolic steroids, unprescribed amphetamine use, etc). Plus there is a distinct lack of adoption of proper nutrition, exercise, and recovery techniques. For example, bases do not have saunas, cryotherapy chambers, or dining facilities with well stocked "healthy" food options. Barring some very elite units who might have a dietician assigned to them or some off-label drug use (amphetamines to maintain alertness during a long flight, for example) the run of the mill soldier, while evaluated for fitness at regular intervals, gets precious little support in actually enhancing performance. Professional sports teams are FAR FAR ahead of the military in this.
So even if there was some sort of genetic physical enhancement program, I suspect you will see it pop up in athletics before the military, at least outside of very controlled societies like North Korea. Of course the development and research of PEDs is severely hampered by the persistent banning of any substance that might give an athlete an edge, so the concept of "fair play" will have to change dramatically in order for proper study into PEDs and enhancement programs in the first place.
What would a free country do with broken or retired super soldiers? We also have a strong respect for free will and choice, could one of these child soldiers opt out of the program? What happens with their genetic enhancements? What are the long term health effects and who pays for the care? I think it would be hard for any type of enhanced human program to justify the cost compared to a drone program, aircraft carriers, guided missiles, etc. Dumping millions into children in the hope that they will be physically, mentally, and emotionally suited for the military is a relative long shot compared to hardware weapons programs.
Plus no government is going to want these guys walking around out of control. Governments already have an uneasy relationship with their veterans, this would be much more unstable if the vets were super-human. Far easier to give a normal person a super-suit that is owned and controlled by the government, so that is where I think we will see more innovation.
EDIT: You have clarified your question a bit. In the broader context of "will genetics make humans better" then yes, I do believe that we will have "super-soldiers" at least from our current POV in the next century. But these will be mods applied ACROSS the citizenry, not concentrated just within the military. The types of hacks useful for military performance are also often hacks helpful for everyone else. Take an ability to go without sleep, enhanced low light vision, or improved hot/cold weather tolerance. These have commercial applications so could be expected to be available to all the citizenry, not just some cloistered "army baby factory".
The problem with creating, from birth, a super soldier is that the development time is just too long and the REPLACEMENT time is crippling. Unless a country can sustain a large standing army of genetic super soldiers (who are apart from the normal population) then it is likely that in a large conflict all of your elite forces will be wiped out early on and will become irreplaceable as there is an 18 year grow time so you are stuck with whoever is already in the pipeline.
This is why technological solutions, like an improved armored vest instead of a grown protective carapace, are preferred. You can ramp up vest production quickly and train normal folks in their use far faster than you can grow bullet-proof soldiers. Plus the vest can get modded with the latest enhancements while the genetic soldiers will always be, literally, a generation behind the current tech. Why give humans the ability to see in the dark instead of developing a better night vision system?
The genetic hacks like a simple, easily understood MHC system (the immune markers that govern self/not-self) to allow for rapid organ transplant from a common inventory would be useful for EVERYONE, not just soldiers. Same with any genetic enhancements what would allow for resistance to disease, radiation, allow for limb regrowth, etc.
An area you WOULD see some military specific interest would be in genetic modifications of ADULTS. We can already do this using viruses to insert gene sequences into cystic fibrosis patients. So a new soldier, once they finish basic training, could get an "upgrade package" of gene therapy that could enhance all sorts of things, like resistance to battlefield biologic/chemical/nuclear agents, possibly ways to reduce traumatic brain injury, methods to alter the immune system to allow for compatibility to a universal blood transfusion or organ transplant system, etc. Boosting SURVIVABILITY to combat is a fairly unique military requirement that may not be necessary for non-combatants, has little disadvantage for veterans (from the perspective of the government that has to deal with them), and would allow for increased combat effectiveness for soldiers independent of their equipment.
Barring a method of being able to brainwash or mind control youth to become perfect soldiers, modifying adults who have already CHOSEN to be soldiers is much more cost effective than raising soldiers from birth with the high likelihood that the majority won't be good soldiers yet can't integrate into the regular society. Even if all you want is a small cadre of elite special forces soldiers, you would need to raise a large contingent of possible subjects, most of whom would lack the mental stability, emotional toughness, and will to be a good special forces soldier, regardless of how physically gifted they are and the specificity of their upbringing. Far cheaper and easier to pull from the regular population and then enhance via gene therapy after successful training.