As long as there's a MOSFET in between, can I supply 24 V bed/extruder heat from an ATX power supply using -12 V and +12 V and ignoring the common rail?
I assume so, I can't think of a reason why not, but want to check.
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No, this is not possible with most ATX power supplies. While in principle you can get a 24V supply by combining the +12V and -12V supplies, the rails are not symmetric, and the negative 12V supply is usually designed for a much lower load than the positive supply. In the example in the following picture, there are two positive 12V rails, capable of sourcing respectively 12A and 15A each, but the -12V rail is only good for sinking 0.5A. If you tried doing what you're proposing, using this supply, you'd be limited to only 0.5A from your "24V" supply.
It is very unlikely that you'll find an ATX power supply with a negative 12V rail capable of sinking significant current since computers don't need such large amounts of current from their negative supplies.
There are also issues with using a MOSFET as you describe. I assume that by MOSFET you are referring to a complete board with various support components, and not just a bare MOSFET by itself. Usually these boards have optocouplers and thus they will probably work correctly, but using just a MOSFET by itself this would not work, as the gate needs to be driven below GND (namely: to below -12V plus the gate threshold) to turn the MOSFET off.