I had this same thing happen on my 08.
Just changed the fuse... there's one for the rear turn signals, another just labelled something like "rear" I think, which is the one that needed changing. It didn't really make sense, because if the rear brake light's working, that fuse shouldn't theoretically have any effect on the rear turn signals, but somehow it did. Anyway it's worth giving it a try... easy fix if it works!
It can get confusing... The brake light circuit and the horn are on the same fuse. The running light circuit, which includes one of two filaments in the tail light bulb and the flashers are on the same fuse. You have 2 filaments in the tail lamp bulb. One filament (running light) is on the running light/ flasher fused circuit. The other filament in the tail lamp (brake light) runs off the brake light/horn fuse. So you have 2 separate fuses that can effect what is happening in the tail lamp bulb. Those 2 separate fuses also feed the the horn and flasher relay circuit.
If you have a working horn but no tail light you know the fuse is good for the tail light and visa versa. If you have running lights but no flasher operation you know the fuse is good and visa versa.
There probably is a broken/disconnected wire/s at the turn signal lamp assembly/s. Grab a 12 volt probe test light. put the 4 way flasher on and find where th signal is lost.
Here's some info I've posted from another thread. It includes original color codes for the wiring of directionals. The tail/brake light filament is a yellow wire and running tail light should be blue.
These colors apply to all FZ6 S1 and S2 models.
Left front turn signal with running light. That's two filament per bulb in the front.
Chocolate colored wire= turn signal filament.
Blue with a red tracer colored wire= running light filament.
Black wire= Ground or negative.
Right front turn signal with running light.
Dark green colored wire= turn signal filament.
Blue colored wire= running light filament.
Black wire= Ground or negative.
Left rear turn signal.
Chocolate colored wire= turn signal filament.
Black wire= Ground or negative.
Right rear turn signal.
Dark green colored wire= turn signal filament.
Black wire= Ground or negative.