Pull up on the block in a drop-top chicken box
A coop is an enclosure or ‘box’ used to contain poultry
“Chickenheads” are girls who like to give head because their heads make an up-and-down motion during the act (like a chicken). See definition #2 here.
So “chicken box” means his drop-top is used to transport chickenheads, not literal chickens or poultry.
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