I have just found exactly this sort of circuit in a small solar powered controller for a garden pond pump.
I am sorry for the messy schematics, this is a small part copied from a larger circuit, reverse engineered directly from the PCB. I didn't really clean it up as I am not done with the board yet. The transistors are some Chinese jellybean parts, in this case they are rated up to 1.5A collector current, but they are not really critical - size them depending on your needs. The battery voltage is around 6V (6V sealed lead acid battery).
Basically pushing the SW1 turns on the Q2 transistor, which turns on Q1 too. Q1 "shorts" the SW1, keeping the entire thing latched on. Pushing SW2 forces Q1 to close by grounding its base, thus Q2 closes too and the load goes off.
EDIT: I have realized only later that the objective was to make the load stop regardless of whether or not the user holds the start button down. The circuit above won't do that as-is. The simplest method is to add a normally close (NC) push button in series with the R6 resistor and remove the original OFF switch. Then when the new switch is triggered, the circuit will turn off, no matter whether the ON button is held down or not.
If you want to use a MOSFET for switching the load, you can either replace the Q2 with a P-channel one or use Q2 to drive a MOSFET - e.g. by putting a resistor to ground in the collector of Q2 and using the voltage drop on this resistor to drive a gate of an N-channel FET.
Disclaimer: This is a reverse engineered circuit, I haven't tried to build it. There could be mistakes in the schematics. YMMV.
Dave has an excellent video on a similar circuit too:
http://www.eevblog.com/2012/03/30/eevblog-262-worlds-simplest-soft-latching-power-switch-circuit/