Using a MOSFET as a switch
The MOSFET is like you say just a switch, and as such will switch on the 4.2 V, but on its own will never make 6 V of it. You'd get 4.2 V with the FET on, 0 V with the FET off.

The DC converter (called boost converter if the output voltage is higher than the input) has an oscillator which also switches a FET on and off, at a high frequency, and the current through an inductor will generate the higher output voltage. Very briefly said.

The question is whether you need the DC converter. You'll need it to get 6 V, but apparently the motor can work over the full range of the input voltage. And then you can do with a FET as switch:

  

FETs are low power: they have virtually zero input current, and can have an on resistance as low as a few 10s of m&Omega;, at 10 m&Omega; even at 1 A gives only a 10 mW dissipation.

The FDC855N can be controlled by a logic level, and has an on-resistance of maximum 36 m&Omega; at 4.5 V control voltage.

If you would rather have a PTH FET then this one is a low cost solution. At 160 mA it will dissipate 0.3 mW!  There are literally thousands of FETs that will suit your needs.