Do I have to use mosfet or I can use NPN such as 2N2222?
It was designed for MOSFETs not junction-transistors. Use the MOSFETs from your VGA card.
The boost voltage is only max to 15v or 24v?
Never let the voltage get above 15V. It was designed for 12V use.
A transistors like the 2N2222 will have about 0.7V on the base to turn it on.
A MOSFET needs about 10V to turn it on. The driver will supply about 10 to 12V.
The bottom MOSFET driver sits on ground and will drive a signal of 0=off, 12V=on.
The top MOSFET is complicated. The "gate driver" part of the IC does not sit on ground. It sits on the output, which is 0V or 12V, switching very fast.
1) When the output is 0V: The driver's power supply is (0V and 12V). The gate voltage is at 0V
2) For a very short time (nS): The driver's output goes high (12V) to turn on the top MOSFET.
3) When the output is 12V: The driver sits on top of the output (now 12V). The driver's supply is on top of the output.
>>>>(0V and 12V)+12V=(12V and 24V)
So "24V" is some thing internal. You should not apply more 15V.