The way I was taught is to combine your breathing and the squeeze. You don’t try to hold the rifle still. You never will. Firstly make sure your rifle is naturally pointing at the target. Look at the target and bring the rifle up and the sights should be aligned. Breath in, breath out and the sights will move through the target - that’s perfectly natural - as the sights come onto the target squeeze in with both hands and the trigger will break as the sights cross the target. Its the technique that was taught for shooting off slings or offhand, and which if using a rest whether standing, kneeling, seated or prone you still use. Also works if the animal is moving. It doesn’t matter whether its a 22lr or a 458 lott your focus is on the target and squeezing and make yourself, the rifle and target as one. It helps if you keep both eyes open and use a relatively low powered scope.
This method probably doesn’t work so well with a heavy barrelled F Class target rifle, bench rest rifle or long distance heavy barrelled sniper type rifle all which will be set up on bipods or adjustable front rests and bags / monopods at the back. These have more than enough weight to control the recoil.
With any rifle, consistency is key to accuracy. Holding the forend and squeezing everything in is one way, but it does introduce quite a bit of human factor. I know that my rifles, if I don’t hold properly will not shoot to point of aim.
The “modern” approach is to remove as much of the human factor as possible and to allow the rifle to recoil on its own straight back. The shooter, generally using high magnification, adjusts the point of aim mostly by slight adjustments on the machine rests. The shot is released with a light touch on the trigger.
Such an approach works and gives amazing accuracy, especially at long range.
But and its a big but, I personally am not carry one of those rifles plus all the kit up the hills or into the woods. Military snipers are young - in their 20’s or 30’s, supremely fit and strong and a 20lb rifle is not an issue. They will set up a static position and most of their job is observation and reporting what they see, but when needed will take a shot at, often, several hundred metres. And they work in pairs with a spotter. Target shooters take their rifles to the firing point in a car, or in several short trips from the car, or use a wheeled trolley.
A hunting rifle is generally a much lighter weight rifle, that is mostly carried. Its job is to put a bullet through the vitals of a deer or other quarry. However we all hunt in different ways, very much depending on quarry and landscape, and even here in Scotland we have huge variation.
Mostly I stalk over ground where 250 would be a very long shot. Most of the ground has enough cover that you can generally get in quite a bit closer, and for me I much prefer spending time getting in close, than spending time recovering an animal shot at longer range. For this I want a rifle and technique that allows me to shoot a buck that is inside a fence in a restock, so it needs to go. I cannot get closer than about 150 as I will loose direct sight of it. So sit down on sticks, bring rifle up, squeeze as steps into the open between two trees and its dead.
An hour later the above buck is in bits in my pack (much easier to carry out across replanted clear fell), and am walking out on a ride through mature conifers with the wind on my back not expecting to shoot anything. A buck steps out at 30 to 40 yards, good solid back drop, cycle the bolt to load a round as I bring the rifle up and shoot it. Sticks still on my pack. Fortunately only 200 yards to the gate.