Regarding the term ἐκπορεύομαι
We have to keep in mind that Trinitarian doctrine and the technical terminology surrounding it did not stabilize until the Fourth Century A.D. The creed commonly called the Nicene Creed would be better termed the “Nicene-Constantinopolitan” Creed, since it incorporates material that was elaborated not only in the first Council of Nicaea (held in 325) but also in the first Council of Constantinople (381).
Although the term ἐκπορεύομαι is certainly inspired by John 15:26, St. John is not using it in the technical sense that the Greek Fathers applied to it (especially St. Basil the Great and St. Gregory Nazianzen). This is important to keep in mind, because the Latin versions of the Gospel of John (both the so-called Vetus Latina and St. Jerome's Latin Vulgate) translated the word ἐκπορευόμενον in that passage with qui procedit. However, the Latin verb procedo is slightly different in meaning from the Greek verb ἐκπορεύομαι (albeit sufficiently similar to be a correct translation when the terms are used in their common, non-technical senses).
I think it would be a mistake, therefore, to infer from the fact that St. John uses the verb ἐξέρχομαι instead of ἐκπορεύομαι in John 8:42 that he is making a theologically precise distinction between his own procession and that of the Holy Spirit. (Etymologically, both verbs mean essentially the same thing: to come from something. See below.)
The technical meaning of ἐκπόρευσις and τὸ προϊέναι
Greek Trinitarian theology makes a distinction between ἐκπόρευσις and something that the Greek Fathers call τὸ προϊέναι.
The difference can actually be seen from the way the terms are formed: ἐκπόρευσις comes from ἐκ (from, out of) and πορεύομαι (to go or come), and thus means “coming from something.” On the other hand, τὸ προϊέναι comes from πρό (forward or before) and εἶμι (to go—not to be confused with εἰμί, to be, which lacks the circumflex accent) and means “going forth.”
Hence the notion of ἐκπόρευσις always contains a reference to the original sender, so to speak, whereas τὸ προϊέναι makes no such reference.
I like to make the following analogy: suppose that the President of the United States sends a letter to the Prime Minister of the United Kingdom. Naturally, he sends it though his secretary of state. Who is the original sender of the letter (from whom does it “ἐκπόρευται”)? The President alone. But who did the sending (τὸ προϊέναι)? Both the President and the Secretary of State.
What can be (and historically has been) a source of confusion, especially between the Catholic and Orthodox Churches, is that the Latin Fathers (for various historical and linguistic reasons) generally fused the two concepts of ἐκπόρευσις and τὸ προϊέναι into a single generic notion that they called processio (from which the English term comes, evidently).
For a good and accessible overview of the distinction between ἐκπόρευσις and τὸ προϊέναι and the Latin processio, see a document called “the Greek and Latin Traditions regarding the Procession of the Holy Spirit.”
Procession in the Holy Trinity
The theological problem that gave rise to Trinitarian dogma is that the Scriptures make it clear that the Father, the Son, and the Holy Spirit are distinct subjects: in the terminology of Trinitarian dogma, distinct Persons or Hypostases.
On the other hand, the Scriptures also make it clear that all three Persons are perfectly and equally God. (I realize that there have been important challenges to this view, but it is the one that the Church Fathers were working with when they were formulating Trinitarian dogma.)
The Scriptures are also unequivocally monotheistic, and the Church Fathers knew from philosophy that God‘s essence or substance must be utterly unique, simple, undivided, and indivisible. To use the technical term, the Son and the Spirit are consubstantial with the Father.
Jesus, whom John identifies as the Divine Word or Son (see John 1:1-15), claims to come from the Father, as we saw, and sent from the Father (see., e.g., John 20:21). How can the Divine Word be consubstantial with the Father and yet also be from the Father? Because the Father communicates His Divine Essence to the Son. (And the Son receives the very same Essence as the Father, entirely, and without division.)
The Greek Fathers (especially those in Alexandria of Egypt) termed this communication of the Divine Essence τὸ προϊέναι.
The Father, through the Son, also eternally communicates His Essence to the Holy Spirit; hence that communication may also be termed τὸ προϊέναι.
The Greek Fathers (in order to combat Manichaeism and similar dualistic systems) made a great effort to emphasize the so-called monarchy of the Father: the fact the He is the unique and ultimate Principle or Cause. That idea is connected with the term ἐκπόρευσις, which concerns these same communications of the Divine Essence, but always refers to their ultimate origin: namely, the Father.
As I mentioned, the Latin Fathers essentially fused the notions of τὸ προϊέναι and ἐκπόρευσις into the single concept of processio, but the same thing can be said: the Son proceeds eternally from the Father alone, and the Holy Spirit proceeds from the Father and the Son (or, equivalently, from the Father through the Son).
(Note that it is incorrect to say that the Holy Spirit ἐκπόρευται—i.e., has His ultimate origin in—the Father and the Son. It is, however, correct to speak of a τὸ προϊέναι of the Holy Spirit from the Father and the Son.)
The short answer
The short answer to the original poster‘s question is that the Son does indeed proceed from the Father (whether you mean ἐκπόρευσις or τὸ προϊέναι), just as the Holy Spirit does. The difference is that the Son receives the Divine Essence from the Father alone, whereas the Holy Spirit receives the Divine Essence from the Father through the Son.
If by “procession” we mean what the Greek Fathers termed ἐκπόρευσις, then we must affirm that the Holy Spirit has His ultimate origin (ἐκπόρευται) in the Father alone. (This is what the Greek version of the Nicene-Constantinopolitan Creed affirms: τὸ ἐκ τοῦ Πατρὸς ἐκπορευόμενον.)
If, however, we mean τὸ προϊέναι, then we must affirm that the Holy Spirit proceeds (πρόεισι or procedit) from the Father and the Son. (This is what the Latin version of the Creed affirms: qui ex Patre Filioque procedit.)
