Uncle Jack's statement is a definite allusion to the Bible, specifically to Matthew 26:39. Enotes give a great definition of allusion in its guide to literary terms:
Allusion - a reference, usually brief, often casual, occasionally indirect, to a person, event, or condition thought to be familiar (but sometimes actually obscure or unknown) to the reader.
Before we address the allusion in this segment of the novel, let's explore the reference in To Kill a Mockingbird in its entirety in regards to the dialogue between Atticus and his brother, Jack:
"You know, I'd hoped to get through life without a case of this kind, but John Taylor pointed at me and said, 'You're It.'"
"Let this cup pass from you, eh?"
"Right. But do you think I could face my children otherwise? You know what's going to happen as well as I do, Jack, and I hope and pray I can get Jem and Scout through it without bitterness, and most of all, without catching Maycomb's usual disease." (Lee 88)
Here we learn that Atticus wasn't truly intent on taking this case: the case defending Tom Robinson. Instead, he was chosen to do so. Atticus admits that this case is already lost because of prejudice: "Maycomb's usual disease." His destiny is already determined. Atticus also reveals that it would be wrong for him to turn away from this impossible challenge. Ironically, it is the same with Jesus who is going through the Agony in the Garden:
And going a little farther, Jesus threw himself on the ground and prayed, "My Father, if it is possible, let this cup pass from me; yet not what I want but what you want." (Matthew 26:39)
Here Jesus is referring, of course, to His Crucifixion. Just look at the astounding parallel! We learn that Jesus wasn't dead set on dying this horribly tortuous death. Instead, He was chosen to do so by God the Father. Jesus knows, however, that his plea is already rejected. His destiny is already determined. Jesus also reveals that it would be wrong for Him to turn away from His Father's will: this impossible challenge.
Further, I find it really interesting that Atticus says, "But do you think I could face my children otherwise?" (88) because although Atticus is obviously referring to Jem and Scout, the same could be said for Jesus. If Jesus had said "no" to His Father, He would be betraying all of God's "children" or His "sheep" that He was supposed to be shepherding. This is one of the many parts of To Kill a Mockingbird that makes Atticus Finch into a true Christ figure.