Does switching gears from drive to neutral while cruising damage your car in any way?

[ChandlerHurlimann]

My step dad, when driving downhill or whenever the car does not need gas, puts the car into neutral. He says it lowers the RPM and saves gas. That's good, I'm curious if that's bad for the car.
Also, I have a Ford Mustang, and to change from drive to neutral, you have to squeeze the button on the right of whatever the shifter is called. is that any different than just pushing it without squeezing the button?
thank you