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?

What you found out is not a solution. Just try to substitute in the equation.
By definition, an equilibrium point is a constant solution of the equation. In this case, substituting $y=c$ leads to $x^2=c^2$ and so there are no solutions (note that by definition a solution is a $C^1$ function defined on some open interval, not on a point).
