People are very quick to emphasize genetics. Mike Tyson did come from good stock but genetics or no, he was a product of his training routine. Do a Google search on Tysons routine. It is incredible. Outside the gym he ran. Inside he did tons of pad work and sparring. What interested me is that most of his strength training took the form of bodyweight exercises rather than weights. 500 press ups plus 2000 situps a day ? That is impressive. I think with enough hard work and consistancy similar approach would deliver outstanding results. It was not just Mr Tyson though. If you view other great fighters they too use a similar training format.
I was answering the question within the context of the original poster. He stated he had an interest in boxing and martial arts. Especailly Mixed Martial Arts. His question is are Squats essential. My argument is no. Useful ? Yes but far from essential. Lifting heavy weights does increase strength. But it's a type of strength that fades quickly after several rounds of fast-paced fighting. Cardio vascular fitness is a fighters prime source of strength. The second you get puffed and winded, you are dead. It does not matter how much you can lift on a barbell, the second you run out of breath you are finished. This is why I have a preference bodyweight exercises done in high numbers. Yes, I reccommend pushups and situps but I use other bodyweight exercises too. The are many combinations that can be used. I can elaborate further depending on how much more you wish to read.
Pushups and situps alone are a terrible way of developing maximal strength.
Alone, they are also a terrible way of developing cardio-vascular endurance.
Whilst I agree with you that there are more important things than maximal strength for a fighter, adhering to a bunch of old boxing propaganda is not the way to go about training. Boxers have always bragged about how many push-ups they do a day because it is something that strikes a chord with everyday vernacular. People know what pushups are.
Check any Tyson training highlight. He is constantly using a barbell. And he had beast numbers in the gym. Is that why he was great? No.
If you want to train a fighter, you need to understand the needs of a fighter, and order them in a hierarchy of sorts.
It might look like this just as an example:
1. Skills.
2. Endurance (many types, aerobic, anaerobic, explosive, repeat explosive etc)
3. Maximum strength
Just because maximum strength comes in as a lower priority than endurance, it doesn't mean that you go about training it with endurance methods. Rofl. That is just inefficient and highly stupid.
You go about training maximum strength in the most efficient way, given the time constraints imposed by the other facets of athleticism you need to train. Any top fighter today trains this way (boxing, kickboxing, MMA, they all lift heavy at some times throughout the year/training camp)