First of all, it is vitally important to realize that our Lord Jesus Christ never said, "You are able to become gods." Taken completely out of context, the phrase from the two parallel biblical texts in question say, "You are gods." This is in the present tense, which makes the Mormon interpretation of this minuscule phrase totally inconsistent because Mormon theology does not teach that human beings who are present here on earth are gods, but rather can become gods in the afterlife. In other words, the underlying reason for John 10:34 being a poor argument for Mormon apologists to use is that Mormonism does not teach that Mormons are gods here on this earth.
John 10:34 is a quotation from Psalm 82, which speaks of divinely appointed human judges, who would "die like men" (Psalm 82:7). In John 10, Jesus Christ was simply pointing to the fact that the Jewish teachers of His time were fatally erring in their doctrine. They were doing so in the same manner as the judges of Psalm 82 erred. Christ was criticizing the grave doctrinal errors of the Scribes and Pharisees who constantly challenged His teachings. He refereed to them as being "whitewashed tombs" (Matthew 23:37). He said that their father was the devil (John 8:44). He rebuked them for teaching as doctrines the commandments of men (Matthew 15; Mark 7). It would therefore be illogical for Jesus Christ to refer to these men as deity. The context of John 10:34 is about the rebuking of false doctrine (John 10:35-37). According to the Bible, there is only one God (Isaiah 43:10-11; 44:6-8), which means that we cannot become gods in the afterlife.
While it is true that Christians will eventually receive new, perfected, glorified bodies in the eternal state, the fact still remains that we cannot become gods. Biblical Christianity has always been a monotheistic religion.