The answer to this question is 'no' for minimal surfaces of revolution in the $3$-sphere:
Consider surfaces in $S^3 = \{\,(z,w)\in\mathbb{C}^2\,|\,|z|^2+|w|^2=1\,\}$ that are invariant under the circle action $$ \mathrm{e}^{i\theta}\cdot(z,w) = \bigl(\mathrm{e}^{ip\theta}\,z,\,\mathrm{e}^{iq\theta}\,w\,\bigr) $$ where $p\ge q\ge 0$ are relatively prime integers. A smooth surface invariant under this action that avoids the circles $z=0$ and $w=0$ can be parametrized in the form $$ \bigl(z(\theta,s),w(\theta,s)\bigr) = \left( \mathrm{e}^{ip\theta-q\phi(s)}\left(\tfrac12(1+u(s)\right)^{1/2}, \mathrm{e}^{iq\theta+p\phi(s)}\left(\tfrac12(1-u(s)\right)^{1/2}, \right) $$ for some functions $\phi(s)$ and $u(s)$ with $u(s)^2<1$.
Such a surface can be shown to be minimal if and only if the curve $\bigl(u(s),\phi(s)\bigr)$ is, up to reparametrization, a geodesic for the metric $$ g = \frac{\bigl((p^2{+}q^2)+(p^2{-}q^2)u\bigr)}{8(1-u^2)}\,\mathrm{d}u^2 + \tfrac14(p^2{+}q^2)^2(1-u^2)\,\mathrm{d}\phi^2. $$ Moreover, the parametrized surface will be an algebraic surface if and only if $u(s)$ and $v(s) = \tan\phi(s)$ are algebraically related.
Now, the metric $g$ is a smooth metric on an orbifold of rotation (the 'poles' of the rotation are the orbifold points, of orders $p$ and $q$ respectively). Making use of the Clairaut first integral, one finds that a geodesic that does not have $\phi$ constant (and hence, does not pass through the orbifold 'poles') and does not have $u$ constant must satisfy a relation of the form $$ \frac{m}{1-u^2}\sqrt{\frac{2\bigl((p^2{+}q^2)+(p^2{-}q^2)u\bigr)}{\bigl(1-u^2-4m^2\bigr)}}\,\mathrm{d}u -\frac{(p^2{+}q^2)}{1+v^2}\,\mathrm{d}v = 0\tag1 $$ for some constant $m\not=0$ with $|m|<\tfrac12$.
Now, it is not hard to show, using Liouville's Theorem, that the first term is not the differential of any elementary function when $p^2-q^2>0$. (Obviously, it is such a differential when $p^2 -q^2 = 0$, but, otherwise, this is clearly an elliptic integral.) Hence, $u$ and $v = \tan\phi$ cannot be algebraically related, since expressing $v$, even locally, as an algebraic function of $u$ would express the first term in $(1)$ as the differential of $\tan^{-1}(v)$, which would be an elementary function of $u$.
Meanwhile, the set of values of $m$ satisfying $0<4m^2 < 1$ for which the geodesic described by the above relation closes smoothly is dense in the interval $[-1/2,1/2]$, since the integral of the first term in $(1)$ over the interval $u^2<1{-}4m^2$ clearly depends on $m$, as it is an odd function that is clearly not equal to zero when $m$ is not zero.
Hence there exist such $m$ that yield a closed geodesic on the orbifold of rotation, which lifts to a closed minimal surface in $S^3$ that is not algebraic.
I would expect that a similar argument can be made for $S^n$ ($n> 3$) using an appropriate symmetry reduction via a group action of cohomogeneity $2$.