Unfortunately, the teaching on 'no soul' lends itself to deliberate misinterpretation by anti-Buddhists, who accuse Buddhists of being materialists.
In fact, the 'existence' of the soul in Buddhist philosophy is denied in the same way as the 'existence' of the body, in that body and soul are both processes, but do not have any static, unchanging inherent existence. It would perhaps be more skillful from a transcultural point of view, rather than to deny the soul, to regard it as a conventional truth like the body.
So what is Milhouse actually buying? A 'thing' or a process?
Punnadhammo Bhikkhu has written an excellent review on this topic:
"Is There a Soul in Buddhism?
To give the short answer first: "No."
As you might expect, the long answer is much more nuanced. The short answer depends on the commonly understood idea of "soul" as an unchanging personal principle that continues in time infinitely. This is the concept of "soul" usually implicit when one begins with the assumptions of a theistic religion. On the other hand, if by soul we mean simply that human beings have a spiritual aspect that is not ultimately bound up with physical processes, then Buddhism would be much more sympathetic to the idea. Buddhism may deny the existence of a "soul" but it is not for that reason "soul-less" in the same way as is materialist philosophy.
Buddhism is often called the "Middle Path." This has been explained in different ways in different contexts. The earliest use of the phrase is found in the Buddha's very first sermon, in which he laid out the "middle way" between the extremes of asceticism and hedonism. On the metaphysical level, the Buddhist doctrine (and more specifically the dependent origination) has been called the "middle way" between the extreme views of eternalism and annihilationism (sassatavada and ucchedavada).
The first sutta of the Digha Nikaya lays out sixty-two false views, or philosophical errors. These make a complex matrix of nuanced positions regarding metaphysical questions but we can simplify them all into two broad categories, (and one additional minor category.) The first major category of error is eternalism, or the belief that there are some "things" (such as a soul) that continue essentially unchanged forever. This was represented in the Buddha's time by all those Indian schools which postulated an eternal "atman", the Self or Soul or "jiva", life-principle. In later times, this philosophy was adopted in some form or another by all the theistic religions like Christianity, Islam or most forms of Hinduism.
The belief in an atman or soul in this sense usually goes hand-in-hand with the belief in a Creator-God, who is the first, most perfect and most powerful of the "souls". Sometimes the soul is seen as a part or a spark of the One Big Soul, as in the Upanashadic idea that Atman equals Brahman. Sometimes the human soul is seen as a separate entity created by God with an act of will. There are other variations on this theme. In any case, the idea of a God as First Principle or Creator would seem to be required once we accept the notion of an essential and eternal soul. The question of where these souls come from can only be answered by tracing them back to a first cause. The inquiry must end in an act of creation by a special ontologically privileged great-soul.
The opposite extreme view is annihilationism, which is a nearly literal translation of ucchedavada (the "cutting-off" view). This, in its simplest formulation, is the view that beings are "cut off" at death and utterly cease to exist. In the Buddha's time this was represented by various philosophies that either postulated the existence of a finite "life-principle" or took a hard-materialist line that denied any separate reality apart from the body.
In western philosophy, this view was developed by some of the stoics and has never completely died out. Today, in the form of so-called rationalism or philosophical materialism it is becoming established as the dominant world-view of the educated classes. On the metaphysical level, it is represented by what is called "physicalism, " the argument that all mental functions are in the last analysis dependent on physical processes. As a corollary, this would mean that such processes are also explicable in purely physical terms, i.e. as specific sequences of firing neurons... Read the rest of the article HERE (highly recommended!)
For a general background see Buddhist Philosophy