What happens when a sine wave goes through very large amplification?

Suppose in my system, I have a signal that is being reflecting back from a sensor, the signal has both amplitude and phase variation as the sensor is moving on the Device Under Test. I want to cancel the amplitude variation of the reflected signal, and someone told me that "Sine wave and huge amplification will results in a square wave with the zero crossing maintained(phase information is maintained)." Can anyone elaborate this further please? If I feed this reflected signal to a very high gain amplifier, how does the amplitude+phase varying signal at the input of the amp look like at the output of the amp?

Your input signal would have a particular frequency and power. As long as both of these are within the specifications of your amplifier, the output signal is just an amplified (sometimes inverted) version of the input.

If the frequency of your input signal is not too high, you can just amplify it to be greater than, say 0.7 V, and use it directly with some digital ICs or rectifiers.

You might want to take a look at power amplifiers (PAs), but your post just mentions sensors and I think a PA is not necessary.

If you need a constant signal level, you can use an automatic gain control:
http://en.wikipedia.org/wiki/Automatic_gain_control

Typically, a signal amplitude grows by amplification and then it gets clipped like in a limiter. If you need only signal phase information, then you can get it from zero crossing. All amplifiers clip growing-amplitude sinusoidal signals.
Most frequency- and phase-modulation systems use amplitude limiters to get rid of unnecessary amplitude variations. If this is the case of your sensor, use the technique.