Nope, you're good. The rules for 5e allow for this kind of paladin all but explicitly. You should have no trouble making and running this character.

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.