Josh is not giving his linguistic prowess enough credit. Han has figured out that 'bro' is a derivative of brother that's popular among a certain subculture of young earth men. He's not just been around him and Chris often enough to have sorted that out, he's seen it among other Earthers in the Nexus.

He just thinks it's going to sound stupid if he tries to say it. Kind of like when he actually tries to speak Shriiwook. Some sounds are just not meant to be made by human vocal chords.

"I gotcha. I'm just saying. Don't expect it from me."