Can nirvana be called as some sort of directed evolution? Can a Buddha or Arahat be considered as an evolved being as compared to the rest?
Of course. In the suttas the Buddha explains how ignorance leads to suffering, suffering leads to seeking the way out, and seeking the way out leads to Nibbana.
So Ignorance, Suffering, Noble Path, and Nirvana are phases of the same sequence. You can call it development or evolution (in the general sense of the word), sure.
Although, to be precise, when we use the word Evolution we usually mean the natural evolution caused by the procreation, crossing, mutation, and natural selection of phenomena (such as the living beings, the memes, the tendencies etc.)
Armed with this knowledge we can describe The Noble Path as Evolution of tendencies under the evolutionary pressure of Karma guided by the Right View.
What are tendencies? Tendencies are mental and behavioral habits that lead to similar thoughts and behaviors in the future. Good habits lead to other good habits. Bad habits lead to other bad habits. The habits or tendencies procreate, they cross-pollinate, they mutate, and they are subject to natural selection. They have all the same forces at play as what we see in the Natural Evolution.
In the nature, the more fit organism can survive and pass on its genes to subsequent generations. In human life, those impulses that we allow to manifest as actions - they too procreate, cross-polinate, and pass their influence to their subsequent generations.
Once you realize that discord is dukkha and clearly see what type of actions generate discord, you reach the conclusion that cessation of unwholesome behavior leads to cessation of discord and thus of dukkha. This point is called The Stream Entry. From this moment onward, those mental and behavioral tendencies that lead to cessation of dukkha have competitive advantage. Your Right View and Right Intention play the role of Natural Selection. Wholesome tendencies procreate and lead to other wholesome tendencies.
This evolutionary process is inclined towards Nirvana. As your habits change and mind clears up, those behaviors that used to generate chaos and discord in your life - go extinct. And those behaviors that lead to peace, harmony, and enlightenment - proliferate and grow stronger.
This process gets more and more refined until it reaches the phase when your own thoughts are the only remaining source of discord. This is when the practice of Jhana is the most beneficial. As you practice the Jhanas, the habits of samatha and vipassana grow stronger, until they become the dominant force of your life. The refinement of these two tendencies culminates in Nirvana.
The scientific definition of evolution is as follows:
Evolution is change in the heritable characteristics of biological populations over successive generations.
There is no guarantee that the offspring of a Buddha or Arahat will be an enlightened being, i.e. these characteristics are not guaranteed to be passed on to their successive generations (Not a very rigorous scientific answer but I hope you got the gist of it).
Another way to think about it is that
Has it been experimentally proved that Nirvana fits into the theory of evolution? OR, Can we design an experiment that can show if Nirvana fits into the theory of evolution?
As far as I know, these questions can not be answered as of now (Perhaps no one has tried to do it). So from a scientific point of view, I would say 'no'.
Evolution in nature perpetuates samsara because it encourages becoming and it encourages reproduction. From the sutta quote below, you can see that going with the flow stands for craving. Craving is the cause of suffering according to the second noble truth. Evolution drives the natural tendency towards craving - sensual cravings and cravings towards becoming.
Going against the flow of nature is renunciation. Trying to end craving is against what evolution in nature drives us towards.
The idea of natural spiritual evolution is not found in Buddhism. The effort to becoming liberated is against natural evolution. Furthermore, as you can see in the Buddhist world today, extremely few join the monastic order in pursuit of liberation.
Natural intellectual evolution only leads towards increasing sensual enjoyment while attempting to reduce suffering, through improvement in healthcare, education, infrastructure, politics and economy.
Natural intellectual evolution does not lead towards liberation from clinging to sensual enjoyment. Rather, it leads one towards asking how to increase enjoyment while avoiding suffering as much as possible.
From Iti 109:
This was said by the Blessed One, said by the Arahant, so I have heard: "Suppose a man was being carried along by the flow of a river, lovely & alluring. And then another man with good eyesight, standing on the bank, on seeing him would say: 'My good man, even though you are being carried along by the flow of a river, lovely & alluring, further down from here is a pool with waves & whirlpools, with monsters & demons. On reaching that pool you will suffer death or death-like pain.' Then the first man, on hearing the words of the second man, would make an effort with his hands & feet to go against the flow.
"I have given you this simile to illustrate a meaning. The meaning is this: the flow of the river stands for craving. Lovely & alluring stands for the six internal sense-media. The pool further down stands for the five lower fetters. The waves stand for anger & distress. The whirlpools stand for the five strings of sensuality. The monsters & demons stand for the opposite sex. Against the flow stands for renunciation. Making an effort with hands & feet stands for the arousing of persistence. The man with good eyesight standing on the bank stands for the Tathagata, worthy & rightly self-awakened."
The internet says:
The theory of evolution is based on the idea that all species are related and gradually change over time.
Evolution relies on there being genetic variation in a population which affects the physical characteristics (phenotype) of an organism.
Some of these characteristics may give the individual an advantage over other individuals which they can then pass on to their offspring.
Buddhism says a Buddha arises from "mere chance" (adhicca), as follows:
Bhikkhus, it is by chance that one obtains the human state; by chance that a Tathagata, an Arahant, a Perfectly Enlightened One arises in the world; by chance that the Dhamma and Discipline proclaimed by the Tathagata shines in the world. - SN 56.48
Therefore, it appears Nirvana does not fit into the theory of evolution because Nirvana or the disposition to reach Nirvana appears not related to ancestral genetics.