OK, well it is pretty clear this is in fact, a bridge arrangement, where "E+" and "E-" are the "excitation" terminals (E+ is controlled by the IC in order not to waste power while the device is sleeping - what trigger turns it on?) and "S+" and "S-" are the "sense" terminals.
I am tying to get my head around it - can you please describe which blacks and whites are connected together in relation to the four red wires?
Oh, never mind, I have it figured. At a guess it must be E+ - blacks - S+ - whites - E- - blacks - S- - whites - E+.
Or E+ - blacks - S- - whites - E- - blacks - S+ - whites - E+, it doesn't matter which.
Ok, that makes perfect sense now, it is indeed a bridge, when you put pressure on the sensors either all the reds come closer in resistance to all the blacks than the whites, or vice versa and that moves one "sense" wire positive and the other negative.
It is going to be pretty difficult to obtain data in a useful from from the COB chip, and it controls the excitation to the bridge so I suppose you will actually have to use only the strain sensors and simply connect the red wires to the Arduino with "E+" going to Aref, "E-" to ground and "S+" and "S-" going to two analog inputs. It goes without saying you will have to calibrate it yourself so there is no real point measuring the voltages in the present device.