As described at https://github.com/petrockblog/SNESDev-RPi#button-polling a shutdown command is issued when the button on the GPIO adapter is pressed five times. Maybe that signal is somehow inserted into the RPi when you connect your controller(s).
The dmesgcommand might give you some more insights into what the shutdown is causing.

Alright quick update.

I went back through the SNESDev github page. https://github.com/petrockblog/SNESDev-RPi#snesdev-rpi

I was able to get my first controller to be recognized and it seems to be working correctly but I had to use putty to ssh in and run sudo SNESDev. If I close putty it stops working.

On the github page it says to run:
sudo ~/SNESDev-RPi/bin/SNESDev &

This fails. I am am not sure what is going on.

I also tried to run:
Running SNESDev as a service

SNESDev-RPi comes with a script that allows SNESDev to be run as a service. The installation command for that is

sudo make installservice

and that fails. I think I am close but I am still missing something here. Can someone who has set this up chime in?