Hund's rule tells us that 1) any orbital in a given subshell must first be occupied by a single electron before being occupied by a second electron and 2) the electrons in a singly occupied orbital all have the same spin.
Diagram I follows both 1 and 2; there are no prematurely-filled orbitals (boxes), and all the singly-filled 2p orbital have the same spin (the arrows are all pointing up). So, diagram I follows Hund's rule and is correct.
Diagram II violates 1: the first box in the 2p subshell is doubly filled while there's still a vacant orbital. That second electron in the first box should be in the third box and pointing upward for diagram II to be correct. Thus, diagram II violates Hund's rule and is incorrect.
Diagram III follows rule 1 but violates rule 2: While every 2p orbital is singly filled, they do not all have the same spin. In the second box, the arrow is pointing down while the arrows in the other two boxes are pointing up. Thus, diagram III violates Hund's rule and is incorrect.
So, of the diagrams given, II and III do not follow Hund's rule and thus are incorrect.
