What characteristics of a MOSFET should I be looking at?
Assume the gate capacitance is 1 nF. For a 4 MHz operating frequency the period is 250 ns so, you might want switching time in one period to be 10% of this (or less) to give decent efficiency. That means 12.5 ns for charging up that gate capacitance or discharging it (maximum).

I = C dv/dt hence I = 264 mA just to charge the gate. 

I took dv at 3.3 volts and dt as 12.5 ns.

Given that you have 40 mA available you'll need a driver because I really don't think you'll find a MOSFET that can handle 10A and have a gate capacitance of 100 pF.