Can quantified renamable Horn formulas be identified using the same procedure as unquantified formulas?
Yes.  The quantifiers can be ignored for the sake of the test since a quantified Horn formula is syntactically identical to an unquantified one except for the quantifiers.  That is, a quantified Horn formula still consists of only of clauses with zero or one positive literals.  The test only determines whether any combination of polarity flips could produce a Horn formula.

In order for the test to be useful for quantified formulas, it only remains to show that a quantified formula can be renamed and remain equisatisfiable with the original formula.

For unquantified formulas renaming a variable leaves satisfiability unchanged because a solution to the renamed formula is the same as a solution to the original formula except that the renamed variable has its value negated in compensation for the polarity change.  The same is true for existentially quantified variables in a quantified formula, and for the same reason.  For universally quantified variables, observe that the universal quantifier already requires satisfiability to survive a polarity flip because any solution has to encompass both true and false values for any universally quantified variable.  That is, $\forall x(x)$ and $\forall x(\lnot x)$ offer the same solution constraint; the clause must be satisfied with $x$ taking both the true and false values.