Hello Matt,
The flag says 武運長久 across the top, which means "everlasting luck in battle". It is the most common phrase found on flags purporting to be from the WW2 era. The rest of the flag is sporting one or two slogans from the era, and the rest are names (Mitsui, Yamashita, etc.). The two red stamps look like post office stamps - maybe post card cancellation stamps? In general it doesn't look like a typical flag that was sent off with a soldier for luck, and I suspect it may be a souvenir item that your great uncle acquired. There is no name or location of a recipient on it.
Can't read anything on the handkerchief.