Mac Mini won't boot. Is it the monitor or the Mac?
The Mac will boot with or without a mouse and keyboard, and you should see something on the monitor either way.
You do not need to use Apple mice and keyboards.

So, the problem has to be either the monitor or the Mac Mini.

You can look at the power light to see if the Mini is running. If it is steady white, the computer is on. If it is pulsing, the computer is sleeping. If it is off, the computer is off.

If the computer is on but nothing is coming up on the monitor:
Check all connections. Try with a different monitor, if possible. Make sure the correct input is selected on the monitor.

If the computer won't turn on:
Check all connections. Make sure it's plugged in. Make sure you've pressed the power button all the way, without holding it down too long.
If you can't get it to turn on at all, take it back to the people who sold it to you and get your money back; they sold you a broken machine.