Connecting and disconnecting clipless bike shoes from Shimano pedals

No, you cannot break the cleats nor the shoes or pedals with any reasonable movements. They are made to withstand even large forces directly pulling the cleats from the shoes and various forces that happen in heavy terrain.

Jest turn your heel outward and that should unclip you easily (at least that's what I do in the MTB pedals made by CB but I believe Shimano is the same, there is no need for any push down hard - correct me if I am wrong for Shimano). Do not try to unclip by pulling the shoe up.

I suspect you're doing this in the air, before fitting the pedals to your bike.

Or possibly you're putting a bare cleat in the pedal.

Either way, its really hard to get things working like that. Instead, fit a pedal to your bike, and install a cleat on your shoe. Then back off the release tension to minimum to begin with if your pedals have a tension adjust.

The length of your shoe should provide enough leverage to unclip.

Those Cleats are practically indestructible! You will be running in those shoes over rocks and gravel in a race! Kick and stomp in and twist out. Way safer then toe clips for MTB!

Most of what you need to know has already been explained in the other answers:

• No need to "push down hard on the heel of the shoe" when unclipping. A simple twist of the foot suffices.
• Adjusting the tension on the mechanism is important. Higher tension means less likelihood of accidentally unclipping, but does mean it's a little harder to get in and out of the pedal. You'll want to balance your concerns of convenience, security, and even crash-worthiness (you don't want your foot staying in the pedal if you fall)

However, there's one really useful piece of advice that I don't see anywhere:

• When clipping into the pedal, you want the pedal position to work with you. If you are stationary, the easiest way to do that is rotate the crank to put the pedal at its lowest point. Then you can just put some weight on that leg and the cleat will clip right in.
  
  Alternatively, I find that I like to get secured as I proceed from a stop. In that case, starting with the pedal just past the high point of its rotation works best for me. The force applied to accelerate the bike is also just right for securing the cleat in the clip.
  
  I typically combine both of these techniques. I get one foot clipped in before I start moving. Then to start, I rotate that foot to the just-past-high point where I can then apply the usual push down to get the bike moving. Finally, as the other side's pedal comes around, that foot is placed onto the pedal and clipped in during its downstroke.

It takes a little practice, but before too long it'll be second nature.