It is possible to shift the square wave a specified amount by using an op-amp and some resistors. But that would not be of any use in this milliohm meter. That is because we don't know a-priori exactly what voltage shift is needed. The goal is to get the square wave to be symmetrical around 1.8 volts, going just as far above 1.8 as it goes below 1.8. A high-pass filter does that very well and very economically.
However there is something that bothers me about this circuit. It looks like it is an attempt to do a classic 4-point resistance measurement. That requires 2 contacts to carry the current to the resistor under test, and 2 other contacts to measure the voltage across the resistor. This removes the contact resistance as a factor because the voltage measuring contacts have essentially no current flowing through them. No current means no offset voltage due to those contacts. But this circuit violates that principle. It shows the bottom current-carrying contact from the resistor going to the ground symbol. It also shows the lower of the two voltage-measuring contacts going the same ground symbol. Therefore there is nothing to prevent the current from the resistor from going through the voltage-measuring contact, especially if that contact resistance was less than the resistance in the current-carrying contact - thus inducing an offset voltage. This circuit does not appear to be a faithful implementation of the 4-point resistance measurement.