First you use $c$ as a parameter, i.e., consider your equation as elliptic curve over $\mathbb{Q}(c)$.
You can also consider this equation as an equation of an elliptic surface $S$. Now one easily proves that $S$ is a K3 surface, and that is the quotient of $E\times E$ by a group of order 2, with $E$ the elliptic curve $y^2=x^3-x$.
From this it follows that the Picard number of $S$ is 20. Using the Shioda-Tate formula it follows that the $\overline{\mathbb{Q}}(c)$ rank of $y^2c(c^2-1)=x(x^2-1)$ is two.
Since this elliptic surface has 4 $I_0^*$ fibers, the torsion group has to be a subgroup of $(\mathbb{Z}/2\mathbb{Z})^2$, therefore $(x,y)=(c,1)$ is a point of infinite order. (Of course you could prove this by showing that $2(c,1)$, $4(c,1)$, $6(c,1)$ and $8(c,1)$ are nonzero.)

Note that $(x,y)=(-c,\sqrt{-1})$ is also a point of infinite order, and that this point is not in the subgroup generated by $(c,1)$.
This implies $E(\mathbb{Q}(c))$ is generated by the four two-torsion points and $(c,1)$. So the point $(c,1)$ is of infinite order and this is the only point you get for free.

For special values of $c$ the rank can be higher. If you were able to control in the way you ask in your question then this would be very great, because it means more or less that you are able to control the rank of elliptic curves under quadratic twisting.

However, there is a weaker results, using 2-descent you can get a bound on the rank depending on the number of primes dividing $2c(c^2-1)$, see e.g. Section X.5 of Silverman's book on the Arithmetic of Elliptic Curves.