Sorry but there are still lots of things that are not clear.
The power source itself is a device that delivers -28V to the power output.
OK but what is it. Is it a battery like you showed or is it some form of power supply derived from the mains. If it is then is the output voltage floating or is it tied to ground.
Output of 7805 to uno input pin,
Is this
a) An arduino input pin?
b) The +5V pin
c) The Vcc pin
d) The power input jack?
uno output pin
Is this
a) A signal output
b) the +5V pin
to a BNC connector that will be hooked up to the downstream device.
So how is this device powered? Is the ground of this tied to any other ground, a BNC plug implies it is. So that when you plug the BNC into your device you will be shorting out the arduino ground with mysterious downstream device.
Is the arduino connected to a computer through the USB at this point. This will act as another connection to the common ground.
It is my guess that this is what is going wrong, that the grounds are all not floating and therefore can't be connected together.