Estimating Torque?

Your area is A = π × 2²

^{12.56 sq inch}

1 pound per square inch = 1/225 * (39.4)² = 6.9 N/m².

= 690kPa

http://www.onlineconversion.com/

Wurst case of 100 pounds per square inch on a 4 inch cylinder. A 4 inch cylinder has about 13 square inches so you will need to exert about 1300 pounds on the piston driving the clay. The motor can be of any size you want (change gearing to get 1300 pounds of force). The question is how fast do you need to drive the clay (volume/min, sec, etc.), that will tell you the power you need.

Have you considered hydraulics?

Or linear electric actuators?

http://www.globalspec.com/search/all?query=linear+electric+actuator&newSearch=new

http://www.amazon.com/Heavy-Duty-Linear-Actuator-Stroke/dp/B012YAY9NM/ref=sr_1_2?ie=UTF8&qid=1459984623&sr=8-2&keywords=electric+actuator

Wurst case? Is this a sausage extruder?

You forget that you are playing to a very tuff house.

And whatever happened to throwing clay? Squeezing it seems so, so gauche.

Google

Conchita Wurst

Perhaps

https://www.servocity.com/html/1570_lbs__thrust_linear_actuat.html#.VwWfs6QrKUk

It depends upon how fast the material is supposed to come out. There is nothing about this in the original post.

Hint: power is torque multiplied by angular velocity.

Radius = Diameter/2 = 2"

Area = Radius^2 x pi

Force = Pressure x Area

Motor Torque = (Thread Pitch x Force)/(Gear Ratio x 2 x pi)

Note: Friction is not accounted for.

Deleted

This 3D clay printer is a wonderful opportunity to exploit the non-Newtonian behavior of some clay suspensions. There is plenty of existing clay research such as
http://iwtc.info/2005_pdf/09-3.pdf

describing the sensitivity of clay viscosity to shear stress and/or shear stress history.
Then take a look at this article
https://en.wikipedia.org/wiki/Non-Newtonian_fluid
Especially look at the 'Bingham Plastic' section, leading to the opportunity to extrude a clay suspension at low viscosity (hence needing less torque due to lower piston force). Such a 'Bingham Plastic' will revert to a stable solid after the extrusion process.
With the right clay suspension, you could gain easy 3D printing and a reliable unfired result.

Mark Bingham
Relativity PL