A few thoughts, some of which will echo what I said to WestKentucky in his thread.
First off, you said the boy is six? You know your son, but six is usually pretty darned small. It is indeed going to be difficult to find a "youth sized" rifle small enough to fit him. Your older son's .243 is very likely quite a bit too large.
(For those of you who don't have kids or haven't had young ones around in a while, six years old is kindergarten or maybe starting 1st grade. It's unlikely that a six year old weighs more than 50 lbs, or is as tall as four feet)
Proper length of pull is going to be right about 8 or 9 inches. That's a very small rifle, indeed.
BCRider was spot-on to say most rifles will be quite hard for him to stretch to hold, let alone see through the scope or over the sights without having the butt kick him mightily, no matter what the cartridge.
That H&R may be just about the closest thing to just right -- specifically because he can whack that butt off and fit a recoil pad that comes close to letting the boy properly address the gun. At least then the kid has a fair chance of absorbing the recoil without being "kicked."
As for the cartridge, I'd say .44 is just fine (yeah, sure, .357 too) BECAUSE you can start as low as you want with the load and get him confident shooting it. Shoot "Cowboy Action" .44 Specials if you don't reload, or make up some light 200 gr. Trailboss 600 fps. loads if you do. You can have it pop like a .22 for him to start out, and once you've shown him how to shoot without getting hurt, you can move him up to something like a Keith 240 gr. SWC load.
You never even have to get into .44 Mag territory. There's just no need. A 240 gr. SWC going about 900 or 1,000 will plow through a white-tail like a freight train, without putting a bruise on Junior's shoulder or making him develop a flinch.