While John 10:22-42 in some ways continues the discourse running before it, the reference to the Feast of Dedication shows that some time has lapsed since the previous discussion, which creates some discontinuity as well. So while there might be a connection to Ezekiel 34, in understanding John's use of the festival, it's better to look ahead to the next exchange between Jesus and the Jews rather than necessarily back to the Good Shepherd discourse. In doing so, the reference to the feast is seen to create a fitting setting for the discussion that follows as well as allow to John to continue emphasizing Jesus' fulfillment of the feasts, and in particular of the temple.
The Feast of Dedication commemorates the re-dedication of the Second Temple following the Maccabean Revolt. According to Wright in "The New Testament and the People of God", the revolt, while successful in some of its aims, left the nation anticipating a Messiah who would overthrow at last the (Seleucid or) Roman rule and establish Israel again as an independent kingdom.
The Feast is therefore an appropriate setting for the question of the people: "How long will you keep us in suspense? If you are the Messiah, tell us plainly." The political overtones of the feast, however, also ensure that Jesus does not answer directly, lest his purpose be misunderstood.
While the Feast of Dedication is not one of the festivals required in the Torah, John also shows that Jesus fulfills this feast as well. Kostenberger argues in "A Theology of John's Gospel and Letters" that John writes a little after the destruction of the temple in 70 AD as (at least in part) an appeal to Jews struggling in the new environment to make sense of their religious practice.
While many at the time turned from a Temple-based worship to a Torah-based one, others still anticipated Messiah's coming to overthrow the occupation and rebuild the temple. John's gospel is then (in part) an appeal to those looking for a new temple to look to Jesus as a new temple. Thus John highlights Jesus' prediction: "Destroy this temple, and I will raise it again in three days."
Continuing this theme, Jesus fulfills the role of the temple in the Feast of Dedication. Whereas in the festival it is the temple that is set apart (dedicated) for God's purposes, both Carson (PNTC) and Kruse (TNTC) observe that John highlights Jesus, in verse 36, as the one "set apart" by the Father.