I'm reading an article by Dan Carmon on square-free values of large polynomials over the rational function field and became interested in the question
What is the use of square-free values? Who 'started' investigating them and why? What was the use (if there was any) back in the day and is there any use now?
I found it very difficult to find extra information this. Any ideas where I should look?
This doesn't really answer the question, but it discusses related matters.
The Mobius function, $\mu(n)$, is defined to be 1 if $n$ is squarefree and has an even number of prime divisors, $-1$ if $n$ is squarefree and has an odd number of prime divisors, and 0 if $n$ is not squarefree. Thus, $\sum_{n\le N}|\mu(n)|$ counts the number of squarefree values of the polynomial $f(x)=x$ up to $N$, so there's a connection to squarefree values of polynomials.
Now, $\mu(n)$ turns out to be a very well-connected function in Number Theory. It shows up in the Mobius Inversion Formula, which says if $g(n)=\sum_{d\mid n}f(d)$ then $f(n)=\sum_{d\mid n}\mu(d)g(n/d)$. And it is intimately related to the Riemann Zeta function, via the equation ${1\over\zeta(s)}=\sum_1^{\infty}{\mu(n)\over n^s}$.
So, if you are interested in Number Theory, then you are interested in the Mobius function, and if you are interested in the Mobius function, you are interested in squarefree values.