This Euler function is essentially the same as Dedekind's eta function (Wikipedia, Mathworld). The usual use of the $\eta$ function is to express various modular forms. In particular, you should be able to rewrite Hermite's solution of the quintic by modular functions in terms of the $\eta$ function.

I don't know whether you can express roots of higher degree polynomials in terms of $\eta$, but I would guess not. Here are my two hazy arguments:

Hilbert conjectured that the roots of a general sextic could not be expressed using functions of one variable. I am told that this conjecture appears in *Über die Gleichung neunten Grades*, Mathematische Annalen Volume 97, Number 1, 243-250; I have not read this article. Abhyankar proves a version of Hilbert's conjecture in this paper, which I discussed in my answer here. Unfortunately, to my limited understanding of Abhyankar's result, it is about *algebraic* functions of one variable, so it is not clear to me that it answers your question.

According to Wikipedia, Hermite's solution of the quintic by modular forms was finally generalized to equations of arbitrary degree by Umemura, using Siegel modular forms. Siegel modular forms are analytic functions of many variables. This suggests to me that a generalization using modular forms in a single variable was found unworkable.

As you can tell by the hesitant style of this answer, I find the results on solving equations by transcendental functions rather hard to follow; they all seem to be written in very equation heavy nineteenth century style. Since these questions come up fairly often on MO, it would be great if someone could recommend a good survey which translates them into the modern language.