1. John 8:34a is the typically Johannine version of the gospels' sayings formula: Ἀμὴν λέγω ὑμῖν. Many scholars argue that this is used to introduce a traditional saying, fewer that it reflects Jesus' own words.  In favor of authenticity is the fact that Amen is Aramaic and is therefore unlikely to have originated in Greek Christian circles.  Moreover, its initial placement is at best odd for an Aramaic speaker.  In both John and the synoptics only Jesus speaks thus, so it is distinctive. Few Johannine sayings introduced with these words have a clear synoptic parallel and even those that do betray no direct dependence on a synoptic text.  So there is multiple attestation of the prevalence of this formula at the oral stages of Jesus sayings tradition behind the gospels.  But this only proves that it is primitive.  It could just as well have been invented by anyone who claimed to speak for Jesus as by Jesus himself.  Here it is used to introduce two wisdom sayings.

2. John 8:34b. Jesus did not invent this observation as there is a clear Stoic parallel in Epictetus 2.1.23 stating that no one who sins is free.  The closest parallels in early Christian sources are in Paul (Gal 3:21-26, Rom 6:16-20).  In neither instance does Paul's argument indicate possible dependence upon a source.  And Paul is more likely than Jesus to be the prime source of Stoic influence upon Christianity.  Indeed, the phrase "slave of sin" is at home in Paul's theology but out of place among the sayings of Jesus, who was himself criticized for associating with sinners.

3. John 8:35. The contrast between the persons who do or do not μένει...εἰς τὸν αἰῶνα is characteristically Johannine, but the pairing of "the son" with "the slave" is not.  In itself the contrast of a son's inherent right to residence with a slave's conditional stay "in the house" is a general observation that is true, regardless of any association with Jesus.  It expresses everyday wisdom within a first-century context.  Hence it is a proverb without any verbal twist like genuine parables and aphorisms to confirm the Johannine claim that it is an offspring of the mind of Jesus.

Yet it is vaguely parallel to the synoptic versions of the parable of the tenants (Mark 12:1-9//Matt 21:33-41//Luke 20:9-16) in three ways:

a. Both are set in a conflict with temple authorities.
b. Both contrast ownership with tenancy.
c. Both mention the lots of "son" and "slave."

But these similarities are found on such a remote level of abstraction that there is no possibility of a direct relationship here between Johannine and synoptic narratives.  The parable of the tenants shows that the setting and type of repartee in John 8:35 was not invented by the fourth gospel.  But this falls far short of proving that 8:35 is based on a genuine Jesus saying.  Even if one grants an originally generic and temporal meaning for the claim that "the son stays (in the house) forever," it is difficult to reconcile this verse with statements about the transient character of human existence which have more reason to be ascribed to Jesus (e.g., "the son of Adam has no place to lay his head").

Recommendation: print John 8:34-35 black (not a formulation of the historical Jesus)