The Knight variant of the Noble background grants three retainers — one of whom is a squire (PHB 136):
One of your commoner retainers is replaced by a noble who serves as your squire, aiding you in exchange for training on his or her own path to knighthood.
The knight is training the squire for combat, but the squire is a noncombatant. They tell you which stats to use (the noble, not the squire for some reason), I'm assuming that is for self-defense situations where the squire is cornered and has no choice but to fight. Leaving direct combat by the squire aside:
You can buy a mastiff, which can be trained as a guard dog or attack dog (I can't recall if it says hunting or attack, but a dog can certainly be trained to defend its master).
Can the knight give the leash to his squire and let him be in charge of controlling the dog?
Specifically to order the dog to protect its master (basically ordering it to stand next to the knight and attack anyone who attacks the knight or the dog itself). I'm assuming here the squire is NOT shouting from a tree top "attack that one, now that one," etc. — he just gives the "go" and "stop" orders at the beginning/end of combat.
Please note, I don't consider "nothing allows this" to be a valid answer if "nothing forbids this," is equally true. If it's a case-by-case, DM-by-DM decision, that's fine.
Best Answer
Yes, but it probably isn't as effective as you might think
Giving the leash to the squire (supposing the dog is loyal to them as well) means that the squire can well give orders to it, to the best of their ability. But the squire is an NPC, loyal to the knight but not directly controlled by the player. So without the knight taking the time to give orders to the squire to be passed along to the dog, it's pretty much up to the DM to play that part.
Depending on the DM and the circumstances, the squire may still be shy of combat and hide away; the dog will be useful to guard both the squire and the knight/group's equipment in the fight. Maybe the squire and dog would be OK fighting against normal opponents but run away from monsters, etc.
Also bear in mind that a dog trained for hunting is different from a dog trained for guard duty is different from an attack dog. Hunting dogs will usually not go after something that could hurt them, guard dogs will stop an intruder or attacker violating "their" territory, but training a dog to pursue and fight an active antagonist is different, and dogs with that kind of training are hard to come by.
House rule possibilities
The core rulebooks don't have rules for exactly how involved a companion animal like this can get in combat, outside of a beast master ranger's class features. If a DM did want to allow something like this and the players really wanted to take advantage of it, it would be a good idea to set some standard ground rules for what the animal is capable of; the temptation is going to be there (in some players) to see this as a set of free extra actions in combat. Every additional attack in a round is a pretty big advantage, so you have to be careful.
Ideally, the effect of this should be somewhere between the effectiveness of fighting while mounted (PHB p. 198) and the ranger's class feature (PHB p. 93). Note that even the Ranger, whose major class feature is dedicated to fighting alongside an animal companion, still has to use their action (or one of their attacks per round, later on) to command their animal to attack or aid, so giving much more leeway than that to a dog that's essentially a piece of equipment can be seen as giving that class short shrift.
Another option might be for the DM to allow the dog to fully take part in combat but to consider it an allied NPC/hireling/henchman...not sure how the players would feel giving the dog a cut of their experience points, but that's a way to allow it free rein.
But this is far into the realm of rulings, not rules, so it's certainly going to vary from table to table.