the factory service manual specifically states to flush it out with clean gear or motor oil, and NOT to use gasoline or kerosene. It doesn't specifically mention brake cleaner, but my hunch is that it's more similar to the stuff they tell me NOT to use than it is to oil. In a conventional diff without any plastic or rubber hoses I have no idea what harm it could do, but I'm not a chemist and it seems like an odd thing to put in the manual if there wasn't a reason for it.
I don't buy the argument that it doesn't matter because it evaporates so quickly, because while it does on an otherwise dry surface, it likely gasses off much more slowly once it's been mixed with residual gear oil. My reasoning for this is that when I have mixed it with gear oil, I can still smell brake cleaner in it hours later.
That said, it's used to clean out probably hundreds of axles per day all over the country and I'm unaware of any failures or damage attributed to brake cleaner.