Whether you get the same waveform or a squared waveform out depends on the biassing and signal levels you use.
If you make it switch hard on & totally off, you get a squared / clipped signal.
If you bias it so it's never fully on or fully off (eg. Class A, but not at high currents) you should get a similar output to what you feed in, just inverted.
That will be the same with either a bipolar or FET device, just with the appropriate bias for each.
For a simple example, use an emitter or source resistor, plus base or gate bias so that has eg. 5 or 10% supply voltage across it (thinking of the transistor as an emitter follower or source follower, at that point).
That part can be thought of as a voltage controlled current sink - the bias (or signal added to the bias) changes the emitter/source voltage, and that voltage across the resistor defines a current, by Ohms Law.
Then use a load resistor from power to the collector or drain, eg. five times the first resistor.
With that set current passing through it, it should always have five times the voltage across it, including any variations.
You have a reasonably linear 5x gain amplifier.
It can be a lot higher gain, but the biassing becomes more critical using the same basic circuit.
The way around that is to use another emitter/source resistor in series with a reasonably large capacitor, across the original one. That has no effect on the DC or low frequency gain, but at higher frequencies the two resistors are effectively in parallel so you get higher AC gain, at eg. audio frequencies.
Or it can just be a capacitor directly across the resistor, but it's easier to get distortion.
>google<
This shows two stages very similar to my description one after the other (so gain 1 * gain 2), hopefully it will help make sense:
(If you want to try something like that, make the base bias resistors at least 10x larger (eg. 47K & 10K or 220K & 47K) and the coupling capacitors bigger as well; it would have very poor low frequency response as shown. And add a decent size capacitor across power and ground).
ps. The load applied by whatever an amplifier stage such as these feeds should be a lot! higher input resistance, compared to the collector or drain load value. 4K7 load on the first stage above, feeding the low value bias resistors on the second stage, is not good and I'd expect distortion and low gain, as shown.
And, the bias resistor values should be adjusted to set the collector/drain voltage somewhere roughly half way between the emitter/source voltage and power.