On the trip up the distance covered by light is larger than the height of the box at rest, on the trip down the distance is smaller. A calculation shows that the whole distance covered in the trip by the pulse is larger than twice the height of the box, which is the distance covered by a light pulse when the clock is at rest.
Since light always travels at the same speed, it follows that the time it takes for the pulse to go the round trip is longer when accelerating than when at rest: clocks slow down whenever gravitational forces are present.
This has an amazing consequence: imagine a laser on the surface of a very massive and compact planet (so that the gravitational field is very strong). An experimenter on the planet times the interval between two crests of the laser light waves and gets, say, a millionth of a second. His clock , however, is slow with respect to the clock of an observer far away in deep space, this observer will find that the time between two crests is larger. This implies that the frequency of the laser is larger on the planet than in deep space: light leaving a region where gravity is strong reddens. This is called the gravitational red-shift (see Fig. 7.9).
As for time dilation, the slowing down of clocks in the presence of gravitational forces affects all clocks, including biological ones. A twin trveling to a region where gravity is very strong will come back a younger than the twin left in a rocket in empty space. This is an effect on top of the one produced by time dilation due to the motion of the clocks. The gravitational forces required for a sizable effect, however, are enormous. So the twin will return younger...provided she survives.