Borrowing ideas from both @Gareth and @xnor:
WLOG one column has at least 3 black squares. Discard the 2 other rows. If any of the 4 other columns has more than 1 black square we are done. We are left with a 3x4 board with at least a pair of white squares in each column. As there are only 3 distinct such pairs but 4 columns we are done.
