What is the distortion introduced by a multiplexer?
ΔX assumes that as the signal gets bigger or smaller there will be a non-linearity. Let's look at that first: -



Let's say you are powering the 4052 from 5 volts (left graph). The change in resistance between -1 volt and +1 volt on your input is about 470 ohms - 350 ohms = 120 ohms. If you have a signal this large AND you are feeding the output of the multiplexer into a low impedance input (say 1 kohm) then yes, you will have a substantial ΔX misshaping of the output.

But, in all likelihood you are using an instrumentation amplifier than has input impedances approaching 1 giga ohm (or at least several Mohms). Let's say worst case 1 Mohm and let's say the nominal resistance of the analogue switch is 400 ohms.

What will the basic potential divider effect be: -

\$\dfrac{V_{OUT}}{V_{IN}}=\dfrac{1,000,000}{1,000,400}\$ = 0.9996 or an error of 0.04%

On top of this is the ΔX error of maybe +/-60 ohms - this will be a worst case error of about: -

\$\dfrac{V_{OUT}}{V_{IN}}=+/-\dfrac{1,000,000}{1,000,060}\$ = +/- 0.006%

However, your input aint +/-1V it's much, much smaller so I'd say forget it. 

The error you DO need to be concerned about is leakage currents in and out of the input pins to the load cell - it has a 1 kohm resistance and the leakage quoted on page 5 is +/- 100nA (worst case).

This gives rise to an error voltage across the load-cell of 1 kohm * 100nA = 100 \$\mu\$V.

Can you live with this? It doesn't sound like you can.

FOOTNOTE

I use DG309 multiplexers for miniscule signals from thermocouples and have no problem with them - however, a thermocouple has a very low source impedance of maybe 1 ohm (or maybe 100 ohms if you include the cable). At ambient temperatures the DG309 exhibit a leakage current of +/-5nA max and this gives rise to a dc error of about 0.5 \$\mu\$V. These will work better than the CD4052 I reckon.