Can I put three NM 12 awg wires into a breaker box knockout?
You can, but you need a 3/4&quot; KO and the correct clamp to do so
If you're working with a 3/4&quot; or larger knockout, you can get 3 12/2 or 12/3 NM cables into it.  You just need a cable clamp that's listed/rated for that many cables; fortunately, the Arlington NM842 is cheap and fits the bill perfectly here.  It can even accept 4 12/2, or 2 12/3 and 1 12/2, or a 12/3 and two 12/2s.