Sadly, I found that the screen
that comes with OS X does not support 256 colors. Starting with this article GNU/Screen with 256 colours in Mac OS X, I built GNU Screen as follows.
git clone git clone git://git.savannah.gnu.org/screen.git
cd screen/src
mkdir -p /opt/etc
mv etc/etcscreenrc /opt/etc/screenrc
./autogen.sh
./configure --prefix=/usr/local/Cellar/screen/3.6.0 --enable-colors256 --with-sys-screenrc=/opt/etc/screenrc
make
make install
brew link screen
Now I can run screen and see that I have 265 colors.