If you have a Path of the Beast Barbarian who is using the Form of the Beast Claws, can this be combined with the Fighting Style: Unarmed Fighting to make the claws do 1d8 instead of 1d6 (if you are not wielding any weapons or shield)?
No, you cannot combine these effects.
Path of the Barbarian's Form of the Beast Claws states
Each of your hands transforms into a claw, which you can use as a weapon if it’s empty.
Emphasis mine. So, the Barbarian's claws count as weapons specifically in this scenario. Unarmed fighting only turns into a D8:
If you aren't wielding any weapons or a shield when you make the attack roll
Therefore, you cannot gain any of the benefits from Unarmed Fighting when you have your claws out. You can either attack with your 1d6 claws twice, or take a different transformation and deal 1d8 unarmed once.