Shifting on an incline:
Stay seated and surge for 2-3 pedal strokes then soft pedal and make your gear change. It doesn't use a lot of energy and it makes up for the resultant loss of momentum during the soft pedal/shift. Indexed shifting and & STI/Ergo has nearly eliminated this skill from most cyclist's tool box. It is possible to develop your skills to do this standing (using brake/shifters) but it requires more skill than the first method I listed. The best cyclists I've ever ridden with knew exactly when, where and how to shift with minimal loss (if any) of momentum. It's truly an art.
To answer the original question, yes a poor rear shift can break a chain. It's basically using one or two of the freewheel/cassette teeth as a cutting tool on the underside of the chain while it's at an angle.