Why is the velocity squared in the lift equation?
Newton's second law is:

$$F = ma$$

The $m$ass in the lift equation is of the fluid's, mass times velocity is the fluid momentum. So the equation can be written as:

$$F = {\Delta}(mv)/{\Delta}t$$

Rearrange that equation and it becomes:

$$F = \text{constant}
\cdot
(m/t)v$$

Mass over time is the mass flow rate:

$$m/t = {\rho}vA$$

So we end up with:

$$F = \text{constant}
\cdot
{\rho}vAv$$

Which is:

$$F = \text{constant}
\cdot
{\rho}v^2A$$

Add $A$ to the constant:

$$F = \text{constant}
\cdot
{\rho}v^2$$



Reference: NASA