Hi, What is the correct (effortless) way to stop the strings from vibrating during rests in a measure with both hands?

Also, when playing power chords my fingers make lot of squeaks when moving from one chord to another even though I do not press down the strings had when moving from one chord to another. Anything I can do about this?
I assume what you mean by stopping the strings from vibrating is you want the noise to stop, just place your picking hand over the strings and turn down your volume and lift you hand off the fretboard if you don't want that sound but that's no big deal, most people don't bother with it
"Music Without Emotion Is Like Food Without Flavour"
Paul Gilbert