BJT Common Collector not swinging "negative"
It is going slightly negative. But you need to replace R3 with 39 ohms or less to develop the full negative half cycle swing.

The problem is the emitter load resistor is only 10k, that's far too high for your load R4. That means it can sink very little current. You need to sink a lot of current to develop a voltage on R4.

To put some numbers on this.

You want your load to go 2.7v negative, which will require 54mA to be pulled out of it by R3 (as Q1 can only source, not sink, current). The R1/R2 divider sets the base to about 5.6v, so the emitter is at about 4.9v. When it swings -ve by 2.7v, it will be at 2.2v. With 2.2v across R3, it needs to sink at least 54mA, ideally a touch more, so must have a maximum value of 2.2/55m = 40ohms, or the nearest E24 value of 39 ohms.

That's the problem with class A amplifiers, which is why we have class B amplifier and totem pole outputs on logic. R3 burns a lot of power all the time to make sure it can output a proper waveform into a low resistance load.