I think "make the vertical height be negative" is a bit simplistic. The key point is to work out what direction you want to measure "y" in (and what your start and end points are).
For the given problem, the following are all fairly reasonable approaches:
Let y be the height above ground, and find out when y = 0. Then y = 20 - gt^2/2 (i.e. gravity provides a negative acceleration, because it's acting downwards, while we're measuring y upwards).
Let y be the distance below from the starting point, and find out when y = 20. Then y = gt^2/2 (now gravity provides a +ve acceleration because it's acting downwards, and we're measuring y downwards).
Let y be the distance above the starting point, and find out when y = -20. Then y = -gt^2/2 (now gravity provides a -ve acceleration because it's acting downwards, and we're measuring y upwards).