Does an equilibrium solution have to be a constant? I'm having difficulty understanding equilibrium solutions. Consider trying to find the equilibrium solution(s) of the following ODE:
\begin{equation}
y'=x^2-y^4
\end{equation}
Firstly, I set $y'=0$ and solved for y:
\begin{align}
0&=x^2-y^4 \\
y^4&=x^2 \\
y&=\pm\sqrt{x}
\end{align}
Now my lecturer/workbook specify that there is only an equilibrium solution when $y=c$, for some constant $c$. This would mean that solutions for $y$ that are functions of another variable are not equilibrium solutions. What then is the meaning of this solution? Just that it isn't one?

As John.B has explained, this equation does not have an equilibrium solution. What you have found is the isocline curve corresponding to slope 0. That is, all points along the curve   $y(x)=\pm\sqrt{x}$ will have zero slope in the slope field of the differential equation.
If an isocline curve corresponding to slope zero, happens to be a horizontal line(i.e. if it happens to be a solution to the differential equation), then that isocline curve is an equilibrium solution.
P.S- Isocline curves are solutions to the equation:  $ y'(x) =f(x,y)= c $
