Thanks for the photo! It was quite helpful!
There are two ways you can connect this DirecTV receiver to your Slingbox 350:
1) Acquire a 10-pin to component cable, and use that to connect the DirecTV receiver (using the 10-pin port we see in the photo) to the INs on the Slingbox. You should be able to acquire such a cable from DirecTV.
2) You can also use an HDMI to component converter to connect the DirecTV receiver to the Slingbox using HDMI (you would then be able to connect HDMI from the converter to your TV as well). Here is a little information on that:
Hope this helps!