I would not throw out any crosses or alike for reasons described in that story.
What is negative energy is subjective. Energy that is negative for humans can be valuable for some trees for example. Also the earth does have a recycling system for the negative energies.

Throwing away an item that supposedly is full of negative energy, does not really sound like a good idea to me. If one would feel the need to dispose of an item to allow the energy to be recycled, in my experience it then should be broken to pieces. But such is rarely needed to be done.