How much force per cubic inch does freezing water exert? What force does it take to maintain it's volume?

If the water can't expand then the pressure in the container will increase, (Boyle's Law) we know that increasing the pressure of the water will lower it's freezing point, you may well be correct, the water may remain in a liquid state.

There are actually a number of solid phases of water, not just the everyday ice we think of.

To clarify, I gather you are suggesting taking a volume of liquid water [at some pressure and temp unspecified] in a perfectly rigid container, then lowering the temp to -40F, then seeing what pressure it exerts and what phase changes, if any, occur.

What you need to do is look for a thermodynamic data table for water at the T & P unspecified above. Look up the density or specific volume.
Then look at the -40F [=-40C which is handy if your chart is metric] and find the pressure at which the density [or specific volume = 1/density] is the same. Look in the tables for water and ice.