Do accelerometers know when they are moving straight up or straight down?
I think an accelerometer and arduino is a great way to count the number of times the box moves up and down. It's a compact, reliable, non-contact way of detecting movement.

It could work for movement as gentle as the rise and fall of your chest, when breathing, lying down, and certainly anything faster than that.

You might get away with a simple threshold detector with some hysteresis - count one when the acceleration exceeds +x, then set the threshold to -x, until it crosses -x, then set threshold to +x.

If the acceleration is more gentle, or there is background vibration which you don't want to count (even though it is actually movement up and down), then the signal might be lost in the noise.
Then you will have to implement a simple digital filter on the arduino, that in effect "searches" for movement at a particular rate. This could be a simple as just adding up the last 100 acceleration samples (at 100 Hz) and putting the average through the threshold detector, or you could design a more elaborate filter.

If you're planning to use an accelerometer for anything, try it out first with your smartphone.

Look on your app store for an accelerometer monitor app, something that records the sensor to a file, and do some experiments. If it makes a CSV file, you can open it in excel and plot graphs etc.

The accelerometer in your phone is quite basic and noisy, but is a good substitute for any other sub-$100 accelerometer, and might give you a rough idea of what a $1000 one could do.

Here's an example of what I've done with the phone's sensor.

Good luck! Post your results here (answer your own question)  if it works out for you.