Negative voltage output Or Controling negative voltage
No, it's not possible, since the voltages and currents involved (12V) are too high.  It will damage your board.

Your arrangement with N-channel MOSFETs is the correct arrangement.

By the way, you are not working with "negative" voltages at all - it's all positive - you just happen to be between the "load" and ground.

This is called a Low Side Switch since it is on the "low" (i.e., between the load and ground) side of the circuit.

The LEDs you have are in an arrangement called Common Anode in that the anodes (+ pins) of the LEDs are all tied together (common) and you control the cathode (- pins) of the LEDs.  It means you can use cheap N-channel MOSFETs that are (as you have seen) easy to control with a microcontroller.  The other arrangement, Common Cathode, where all the cathodes (- pins) are tied together and you switch the power to the anodes (+ pins) requires P-channel MOSFETs and N-channel MOSFETs to switch the P-channel MOSFETs, which increases costs.