I encountered a mental block in my understanding of natural units. According to natural units, $c = 1$, which would imply that if we wanted to convert velocity from SI units ($m/s$) to natural units (dimensionless), we would use the relation:
$$c = 1 \implies 3 \times 10^8 m/s = 1$$
Thus,
$$1 \frac ms = \frac 13 \times 10^{-8}$$ (in natural units)
Now, if in a given problem, the velocity is $v_{SI} = 220 \times 10^3 m/s$ (same as $220 km/s$), then the corresponding value in natural units should be:
$$v_{nat} = 220 \times 10^{3} \times \frac 13 \times 10^{-8} = 0.000000733 = 7.33 \times 10^{-4}$$
If I am asked to square the velocity ($v_{SI}^{2}$ or $v_{nat}^{2}$), then I seem to arrive at a problem.
$$v_{SI}^2 = 4.84 \times 10^{10} (m/s)^2$$
Converting to natural units, $$\frac{v_{SI}^2}{c^2} = 5.38 \times 10^{-7}$$
which I also obtain if I square the velocity in natural units,
$$v_{nat}^{2} = 5.38 \times 10^{-7}$$
However, I find this counter intuitive, because while working entirely in conventional units (SI, CGS, MKS, etc.), the square of a velocity ($v_{SI}^{2} = 4.84 \times 10^{10} (m/s)^2$) would yield a larger value of the velocity, whereas in the natural units, it yields a significantly smaller value. Am I doing something wrong, or have I got the intuition wrong? Is this an intended effect of utilizing natural units, and if so, how does this justify effects we observe?