How do I get a backslash when in Japanese input mode in MacOS?
Turns out there's a setting in Keyboard Settings. Change "¥" key generates to \ (Backslash)
No idea why Yosemite changed that setting from whatever it used to be or if it didn't exist before why they felt the need to change the behavior.
Google's Japanese IME has a similar option
I found an answer here and again here
If you hold down the "alt option" key on your MacBook keyboard and then hit the ¥ key, it will type a backslash.