In your imagination, of course it does. But, as you know, John's gospel is focused on Jesus' theology of Himself as the Son of God. One aspect of this is the Good Shepherd.
He is specifically referring to the Pharisees here as "false" shepherds, who attempt to usurp His mission with their legalism and false claims. He uses the imagery of a thief sneaking over a wall to illustrate that the Pharisees cannot be true shepherds as they cannot claim to be the fulfillment of the prophecies, which prophecies allow Jesus to "enter by the sheepfold gate."
To say that this also refers to illegal immigrants is a stretch, and would not have been understood by anyone who heard Jesus' words here.
But, you are free to believe whatever you want.