I contacted the support at eStop and they gave me the following advice.
Analog joysticks are typically potentiometers that are mechanically
centered. Nowadays there is no trim adjustment to them. Luckily there is
a calibration procedure that can be done within the windows environment
that can rezero the analog joystick settings. On my windows 7 desktop,
this can be found in the control panel->hardware and sound->devices and
printers, menu. This is where the eStop Robotics HID will show up as a
device. When you right click on it and select GAME CONTROLLER SETTINGS a
window will pop up. Select the eStop Robotics HID and then click on
properties, in the new popup window, under a tab called settings, select
the calibrate button. This will start a wizard that will walk you through
calibrating the analog inputs to the CCI. If you aren't using all the
analog inputs, that's ok, just click through the wizard and calibrate what
you do have attached. This should take care of your joystick calibration
and give you the correct zero.
The calibration wizard solved my problem. It also works on Windows 8, though I'm not sure about with Windows 10.