Yes you should be strength training. Proper strength training in the major compound movements such at the barbell squat, deadlift, and press will train your CNS to produce more force and increase the strength and stability of your shoulder girdle, core, back and hips. This is how you produce force in the punch and kick.
Some people claim that strength training will make you a slower boxer and decrease your flexibility. For one, I think all the evidence of sports science shows that the stronger you are, the more force you can produce. Technique is important and you need good technique, but being stronger will allow you to strike that much harder after your technique is good. If strength training made you slow, then why are NFL players fast? It's common sense. As far as deceased flexibility, if you're squatting to full depth you should be pretty flexible. Your joints should be going through a full range of motion. You should definitely be flexible enough for what hiding requires.
That's two different situations. For one, if they continued for a while their bodies could adjust. Second, they have no real need to produce force. Moving your arms gracefully is all the way to the left on the speed/strength curve while boxing is closer to the middle. Producing force in sports is all the same. You need to get stronger to produce more once technique is solid.
Sorry but I gotta expand a little on this one. While it's certainly true that a stronger athlete is a better athlete, the force velocity curve is an extremely simplified way of understanding force production, power and speed. Sure there's some correlation between max strength and power, but when it comes to speed it's a lot more questionable. Power as defined here is the ability to produce a lot of force against significant resistance. Speed is the more important attribute when it comes to boxing. What the research really shows it that many factors influency why, and how, someone is explosive and/or fast. Max strength, or maximum force production, seems to be one factor, but beyond novice/intermediate levels they look to be marginal. Other factors like contractile velocity, RFD, mechanics, motor engrams seem to be more important and strength adaptions seem to be highly velocity specific. Meaning that it's not just about how much force you produce, but how fast you can produce it, and that the correlation is not always linear.
You see this in a clinical setting as well. Some people are very strong on the compounds, but can't produce force quickly and are slow as shit. However, some people are also very strong on the compounds and can produce a lot of force quickly and are fast. I'm not arguing against doing strength training here, I'm pointing out that max strength doesn't necessarily correlate with increased velocity. Explosive characteristics are much harder to train than max force production.
The reason why NFL players are as fast as they are are 1) Genetics. Fiber type characteristics, neurological efficiency, contractile velocity and properties, inner moment arms, so on. A lot of this is highly genetic. Genetics is the number one by far, they were born fast and they are build for speed. 2) Training sprinting and running technique and doing the run thousands of times. 3) Doing strength/explosive/plyo training. However, running fast as a big dude with gear on is a very different from having fast hands. You can look at some of the strongest guys in MMA, they are not necessarily faster than the others (even the ones that directly came from the NFL). Max strength definitely helps more in grappling though, compared to boxing.
If you want to stick with the force-velocity curve, boxing is a sport almost as far to the right as you can be.
Another point I want to drive home is that while compounds gives you certain stability elements, they don't really stabilise your joints as much as you'd like to think. Without getting too technical, as soon as you stand on say one leg, the stability requirements of your glutes and core are different. You have to stabilise against your hips shifting and rotating, all elements that you don't train during regular compounds. In regards to the upper body, lifting heavy also highly recruits something called the global mobilisators, which are the stronger muscles mostly responsible for big movements. They do stabilize to some degree, but smaller muscles like the rotator cuff, delts (especially medial), serratus, trap3 and others simply don't get a lot of work in most people. You also don't have the neurological element of stabilisation, which is in some regards a skill.
On the mobility side, while it's true that a deep squat (not everyone can or should hit very deep squats, including athletes) gives a stretch, and other forms of weightraining actually have a good impact on mobility, there are several muscles and joint movements you don't get mobilised. There's no significant internal or external rotation of the hips, no adductor mobilisation, no abductor mobilisation really and so on. Muscles, and especially fascia, likes to be moved about and need that in order to move efficiently. Some might have really good mobility in general, but basic compounds, while being a good addon, simply don't address it nearly enough if the person doesn't.
If you want to talk about force production, good joint movement is key. I wrote a short, but detailed, piece on some of the elements of power production in a punch if you want to check it out:
http://forums.sherdog.com/posts/125416715/
I also highly suggest checking this vid out for anyone in any rotational sport (which is what boxing is afterall):
In the end, I'd like to reiterate that I believe any athlete should do a proper S&C program and that it helps performance a great deal. Compounds being a part of that program. I just feel it's important to be realistic about the limitations of said lifts, and that after the point of diminishing returns you'd be better off using other training modalities. Also it's important to note that different sports require different programs and you'd want different attributes, to a certain extent.
A well rounded, planned out periodized program suited both for GPP, and the sport, is the best option.
