Considering the Catholic Church teaches that there are three persons in one God (St. Patrick famously used the three-leaf clover as an analogy), there have to be at least three persons.

I'm not sure what religion your friend is, but he or she doesn't follow the canon of the Catholic Church.

In addition to the above great answers, I would also like to quote the Fifth Lateran Council, session 8, which for me also seems to oppose such claims:

Consequently, since in our days (which we endure with sorrow) the sower of c**kle, the ancient enemy of the human race, has dared to scatter and multiply in the Lord's field some extremely pernicious errors, which have always been rejected by the faithful, especially on the nature of the rational soul, with the claim that it is mortal, or only one among all human beings, and since some, playing the philosopher without due care, assert that this proposition is true at least according to philosophy, it is our desire to apply suitable remedies against this infection and, with the approval of the sacred council, we condemn and reject all those who insist that the intellectual soul is mortal, or that it is only one among all human beings, and those who suggest doubts on this topic. For the soul not only truly exists of itself and essentially as the form of the human body, as is said in the canon of our predecessor of happy memory, pope Clement V, promulgated in the general council of Vienne, but it is also immortal; and further, for the enormous number of bodies into which it is infused individually, it can and ought to be and is multiplied.

http://www.papalencyclicals.net/Councils/ecum18.htm

What this "person who has made a few interesting claims" appears to be arguing is the Averroist tenet that there is one collective soul for all humans (i.e., that humans do not have individual souls).

According to the OED, an Averroist is

One of a sect of peripatetic philosophers who appeared in Italy some time before the restoration of learning, and adopted the leading tenets of Ibn Roshd or Averrhoes, an Arabian philosopher born at Cordova, viz. that the soul is mortal, or (as others stated it) that the only immortal soul is a universal one, from which particular souls arise, and into which they return at death.

The Church has vehemently opposed Averroism in part because (1) Catholics must believe that God creates a unique soul for each human (a human is an unique soul + body), and (2) Averroism destroys all moral responsibility for an individual's actions.

For an excellent refutation of Averroism, see St. Thomas Aquinas's short work De unitate intellectus contra Averroistas (On the uniqueness of intellect[ual soul] against the Averroists).

Objection 1: Other people are only another physical form of yourself.

Objection 2: There exists only one unposseded [by ego] and indivisible mind.

No, the Catholic Church doesn't believe in this. In Matthew 26:28

for this is my blood of the covenant, which will be shed on behalf of MANY for the forgiveness of sins.

Jesus wouldn't say for "many" if there was only one soul/self. The whole idea of salvation wouldn't really make sense if there is only one soul experiencing all these lives in different bodies.

Objection 3: For anyone, who wishes to think reasonably and wisely, [the one unpossesed and indivisible mind] is the mind of Holy Spirit. For those who prefer to let themselves be possessed by a separate mind (located in the brain – egoistic ego), Holy Spirit is inaccessible and their thinking by the so-called brain is, euphemistically speaking, a dissimulation of thinking.

This statement totally contradicts our free-will. http://www.vatican.va/archive/ccc_css/archive/catechism/p3s1c1a3.htm

Freedom is the power, rooted in reason and will, to act or not to act, to do this or that, and so to perform deliberate actions on one's own responsibility. By free will one shapes one's own life. Human freedom is a force for growth and maturity in truth and goodness; it attains its perfection when directed toward God, our beatitude.

The O.P.’s question is not addressed by Church documents, because it is a philosophical one. Nevertheless, based on Church teaching, it is possible to assess the position expressed by the interlocutor.

What is a person?

In order to understand this question, it is necessary to understand what the Church means by “person.” Although the Church does not make a definition in Magisterial documents, the term does have a history in Catholic dogma:

• The term “person” is used in Trinitarian theology. The Father, Son, and Holy Spirit are Persons or Hypostases (Catechism of the Catholic Church [CCC] 252).

• It also appears in Christology: we say that the two natures of Christ are united according to the hypostasis; or, that Christ is two natures in one Divine Person (the Son) (CCC 466-468).

The most commonly accepted definition among theologians of the term “person” is the one given by Severinus Boëthius in his Liber de persona et duabus naturis:

Persona est naturae rationalis individua substantia [A person is an individual substance with a rational nature] (III, PL 1343D).

By “individual substance,” Boethius means any single entity that stands by itself (things like trees, stones, human beings, and angels; but not, say, the color of the tree, the hardness of the stone, the intellect of the human being, or the actions of the angel). An individual substance is a person, however, only if its nature is rational: that is, if it is capable of truly intellectual knowledge. (In the list of substances above, only human beings and angels qualify as persons.)

Is there one person, or more than one?

Based on this definition, it is clear that there is more than one person.

Even the Persons of the Holy Trinity are really distinct from one another (CCC 252). Since God has created angels and human beings—and evidently multiple individuals of the human species—He has, moreover, created more than one person.

Clearly, therefore, the interpretation given by the O.P.’s interlocutor is very much contradicted by Church teaching.

(When Christ prays that we be “one” in John 17:21, he is clearly referring to a moral, not a metaphysical, unity.)

No, in fact the Catholic Church teaches no such thing. One approach to seeing this is to understand the way the Church sees the body and soul:

The human person, created in the image of God, is a being at once corporeal and spiritual. ... The Church teaches that every spiritual soul is created immediately by God ... and also that it is immortal: it does not perish when it separates from the body at death, and it will be reunited with the body at the final Resurrection.

(Catechism of the Catholic Church paragraphs 362, 366)

This implies that every human being is a separate creation with an individual soul, so that "Other people are only another physical form of yourself" is in fact false. So, indeed, is the statement that "The body we don is temporary". Our bodies, though they will die, will return at the Resurrection where they and the soul will be united eternally.

This doesn't say much directly about the mind. Aquinas, however, did deal with this. In his Summa Theologica, First Part, Question 76 Article 2, Aquinas asks "Whether the intellectual principle is multiplied according to the number of bodies", that is, whether there is just one mind, or one mind for each body. His conclusion is:

If there is one intellect ... in no way is it possible to say that Socrates and Plato are otherwise than one understanding man. And if to this we add that to understand, which is the act of the intellect, is not affected by any organ other than the intellect itself; it will further follow that there is but one agent and one action: that is to say that all men are but one "understander," and have but one act of understanding. ...

However, it would be possible to distinguish my intellectual action from yours by the distinction of the phantasms---that is to say, were there one phantasm of a stone in me, and another in you---if the phantasm itself, as it is one thing in me and another in you, were a form of the possible intellect; since the same agent according to divers forms produces divers actions. ... But the phantasm itself is not a form of the possible intellect; it is the intelligible species abstracted from the phantasm that is a form. Now in one intellect, from different phantasms of the same species, only one intelligible species is abstracted; as appears in one man, in whom there may be different phantasms of a stone; yet from all of them only one intelligible species of a stone is abstracted; by which the intellect of that one man, by one operation, understands the nature of a stone, notwithstanding the diversity of phantasms. Therefore, if there were one intellect for all men, the diversity of phantasms which are in this one and that one would not cause a diversity of intellectual operation in this man and that man. It follows, therefore, that it is altogether impossible and unreasonable to maintain that there exists one intellect for all men.

This is difficult; but essentially what he's saying is that if there were only one mind, then there would be only one "understander" and therefore only one way of understanding each thing. Everyone, then, would understand everything the same way. But this is obviously false. Therefore there is not "only one unpossessed and indivisible mind".

What your friend seems to be embracing is not Catholicism, but something a bit like Buddhism and a bit like some form of gnosticism.