The same asymptotic is conjectured to hold for arbitrary $f$, reducible or not. Following an idea of Granville, it is possible to prove that it follows from the ABC conjecture. (See section 12.2 of "Heights in diophantine geometry" by Bombieri and Gubler). However, even the existence of infinitely many square-free values of $n^4 + 1$ is not known unconditionally. On the other hand, using an elementary sieve argument, it is possible to prove the conjecture for the case when $f$ splits into factors of degree at most 2. (And, I believe, even of degree $\leq 3$, although this is more difficult.)

Indeed, I intended to write $y^2 = x^3 + x^2$ (for a nodal rational curve). Thanks for the correction!

By the way, the nodal curve ${y^2 = x^3}$ is not simply connected, so the result does not extend to singular rationally connected varieties.

Yes indeed; the full proof of the classical simple connectedness of a complex rationally connected smooth projective variety can be found in Chapter 4 of Debarre's book.