How do you say...

How do you say forty four in Japanese?

44? shijuu-shi or yonjuu-yon...
The latter is commonly used today.
In some cases, maybe the former sounds a bit formal/aged and children use it too little, but both make no difference meaning.

What about shijuuyon or yonjuushi?

If I remember correctly, I have heard someone said shijuu-yon.
In some cases, some use shijuu-yon...
It sounds a bit rustic/aged, I think?

I can't say that both are mistaking,
but they are a bit strange and a bit hard to hear/listen/catch.
Maybe they are understood as a slip of the tongue because of the lack of a sense of unity.

What about yonjuushi?

Yes you do say yonjushi without counters like when you count up. It goes yonjusan, yonjushi, yonjugo..,though many prefer to say yonjuyon when count down.
Personally, I've never heard people saying sijuyon.