Electrical – Positive and Negative Mux input in ADC

Some observations:

1. The two middle opamps don't have power. Opamps need power to function according to spec.

2. Actually after Steven's comment I see all 4 opamps are in a single package, which apparently is powered. The problem is therefore one of bad schematic drawing. When you have multiple parts in a single package with common power like this, usually you show the power pins as a separate schematic part, like U13E for example. You can show the power and ground pins on one of the four opamps, but that can add clutter and makes that amp look more special than the others. However, never do what you did, which is to show the power and ground pins on each schematic part.

   Even worse, you then showed two of the parts with power and ground connected, and two without. What if the two pairs were connected to different power and ground voltages? That would be really misleading since it's not physically possible. This kind of schematic convention makes no sense and invites exactly the kind of confusion shown in point 1 above. I left that there so you can see what the first reaction to this misleading schematic was.

3. There is no bypass cap anywhere accross the opamp power pins.

4. You don't say what voltage Vcc is. The TL084 needs a few volts headroom from each power rail, so Vcc should be at least the largest signal voltage plus the headroom.

5. 220 Ω seems very small for R33. Can the TL084 really drive that much current (I haven't checked)? Even if it can, why is that necessary?

6. With D13 in series with R33, Vdetect will be undriven (high impedance) when the opamp goes low. Is this really what you want?

7. D17 shouldn't be right on the output of U13C. That will violate the max output current source capability of the opamp when it is trying to drive high.

Here's a single supply inverting opamp configuration that will do what you want. You will need an opamp capable of output drive to it's lower rail (You will probably want to include a small capacitor across R2 to limit bandwidth, since you don't need much for thermocouple readings)
R3/R2 may need to be increased in order not to load thermocouple depending on type - EDIT, just noticed the output is coming from the AD595, so it's probably low impedance (not checked datasheet) and fine as is:

R3/R2 simply divide the input voltage by 2. R1 and R5 present 400mV to the positive input. Since the opamp tries to keep the two inputs equal, it creates a level shift. For example, when there is -1.2V at the input, to keep the inverting input at 400mV, there needs to be 1.2V at the output. We can now see R3/R2 as a voltage divider with -1.2V at one end and +1.2V at the other, we get 2.4V across R3+R2, so the voltage across R3 is:

2.4V * (R3 / (R2 + R3)) = 2.4V * (10kΩ / 15kΩ) = 1.6V and so:

-1.2V + 1.6V = 400mV

You can run the calculations for the other input voltages and see how it works across the range (remembering there is always 400mV at the inverting input, and effectively no current flows into the input)

Another way to look at it given the above is, say we have -0.6V at the input. We know there must be +0.4V at the other side of R3, so the current flowing through R3 is:

(0.4V - -0.6V) / 10kΩ = 0.1mA

Now we know none of this current flows into the inverting input, so it must flow through R2:

5kΩ * 0.1mA = 0.5V

0.4V + 0.5V = 0.9V at the output

Simulation:

If you need it non-inverting, you can easily do this in firmware or add a simple inverting buffer after this.

ZIGBEE ADC

Just had a look at the Zigbee datasheet and it seems the Vref is fixed at 1.2V (although there is Vref pin, I couldn't find any mention of how to use it in the analog IO section), so you have to work with this unless you use an external (possibly higher resolution) ADC and feed the data to the Zigbee. It's a 10-bit ADC, so 1.2V / 1024 = ~1.17mV LSB, which won't be so bad with with filtering (which use a low cutoff since you have a slowly changing signal from the thermocouple)
Bear in mind the ADC595 has an calibration error of around +-1°C (or +-3%deg;C depending on which variant you are using) so absolute accuracy will not be excellent, but you could go for a higher resolution as mention if you wanted to.
So read the ADC595 datasheet advice thoroughly, pay attention to the PCB layout (if possible a 4-layer with solid ground plane), keep any digital signals away from the analog as best you can and use plenty of decoupling and all should be well.