Yes, Flicker will allow the creature to return to the battlefield, and it won't be exiled at the beginning of the next end step.
Unearth sets up a replacement event: "If it would leave the battlefield, exile it instead of putting it anywhere else."
To me, there is some ambiguity in the english here. Does the "anywhere else" refer to "anywhere other than the battlefield" or "anywhere other than the exile zone"?
I've thought about it, and I've come around to the other point of view. If it meant "anywhere other than the battlefield" then "exile it instead" would have sufficed. The Shards of Alara FAQ and Wizards Community Wiki makes the point that exile effects still happen, even to a creature "unearthed".
Since my conclusion is the same, but the path taken, different, I shall revise this answer accordingly.
Oracle text for Flicker:
Exile target nontoken permanent, then return it to the battlefield under its owner's control.
So "Exile" means the creature will leave the battlefield. Because of the above interpretation of the Unearth keyword's rules text, this event will go on unmodified, as the exile zone is where the creature was heading.
400.7. An object that moves from one zone to another becomes a new object with no memory of, or relation to, its previous existence. There are seven exceptions to this rule:
400.7g A resolving spell or activated ability can perform actions on an object that moved from one zone to another while that spell was being cast or that ability was being activated, if that object moved to a public zone.
Since the Exile zone is public, the second part of the resolving Flicker spell still happens due to 400.7g. It hasn't "forgotten" that the creature in the exile zone is the same one that was on the battlefield. The creature returns to the battlefield under your control.
There is no exception to 400.7 for Unearth, it lost track of the creature when it was exiled. Therefore, it will not be exiled at the beginning of the next end step. Flicker has allowed it to live on.