Why would GDP X GDP Per Capita be used as a measurement of power? What's the purpose of squaring GDP?

From the text, "GDP per capita" is used as a very rough proxy to efficiency. The idea seems to be that if you have a lot more GDP per capita than average then you're probably also getting better potential military value per dollar of GDP than a country where most of it goes to subsistence or other expenditures that aren't optional and hardly contribute anything to a war effort.

Another way to view this metric is as Population * (GDP/capita)^2. That is, a statement that military effectiveness per unit population increases superlinearly with economic development. Something like this probably does hold, though not necessarily with specific exponent 2.

I think the thought is that:

•  using GDP alone biases towards countries with large populations (like China) which might not have the level of resources implied by a high GDP as their large population drains much of that wealth. 
• On the other hand, using GDP per capita biases towards very small rich countries, that may not have the power implied by their high GDP/cap, as their size limits how powerful they can be (e.g. Singapore). 

So, you instead use a measure in between the two, and the naive way of doing this is by multiplying the two measures.  One way of thinking about it is that GDP represents the total resources available to a state, while GDP per capita is a rough measure for how efficiently those resources can be put to use.

Another intuitive way of thinking about this is the surplus domestic product (SDP), which if I recall correctly results in similar rankings as GDP * GDPPC.