Sentence: Anyone who helps all those who help her is helped by all those she helps.
(∀x)(x helps all those who help her ⊃ x is helped by all those she helps)
I paraphrased the antecedent, by itself, as: (∀y)(y helps x ⊃ x helps y)
I paraphrased the consequent, by itself, as: (∀z)(x helps z ⊃ z helps x)
Using 'Hxy' to symbolize 'x helps y' or 'y is helped by x', I then put these together as follows:
(∀x)[(∀y)(Hyx ⊃ Hxy) ⊃ (∀z)(Hzx ⊃ Hxz)]
But now I am wondering if I can instead write:
(∀x)(∀y)[(Hyx ⊃ Hxy) ⊃ (Hxy ⊃ Hyx)]
I am not sure this is correct but any help would be greatly appreciated. This is practice question IIIAi from Goldfarb's Deductive Logic.
Your translation of the two sides of the implication is correct, but in your final result you are mixing up the z and x in the consequent (presumably just a typo), so your translation should read
(∀x)[(∀y)(Hys ⊃ Hxy) ⊃ (∀z)(Hxz ⊃ Hzx)]
This formalization is correct, provided that "she" is meant to be a gender-neutral generic pronoun that refers back to the individuals quantified over by ∀x. When I first read the English sentence, I thought that "she" referred to some unnamed third person. If that's what is meant, it should be treated as a constant, such as $s$:
(∀x)[(∀y)(Hys ⊃ Hxy) ⊃ (∀z)(Hsz ⊃ Hxz)]
That's an ambiguity in the English language and which one is correct depends on what the author had in mind; but I would assume that at the time the book was published, gender-neural "she" was not yet as widespread and the version with "she" as a third party is the intended solution.
As for your actual question: No, the second formalization is not equivalent. It implies the first, but not vice versa; i.o.w, the second formalization is more specific and is not true in all situations in which the first one is.
For a counter model, take a domain with three objects and a circle-shaped helping relationship: $a$ helps $b$, $b$ helps $c$ and $c$ helps $a$.
Then for each of those individuals, the antecedent in the first formalization is false, because they are not helping the only person who helps them, and since the antecedent is false, the main implication is true for all $x$ and thereby the first sentence holds in this structure.
However, the second sentence is false: With e.g. $x \mapsto a$ and $y \mapsto b$, the antecedent is true because $Hyx$ is false, but the consequent is false because $Hxy$ but not $Hyx$. Since not for all pairs of $x$ and $y$ the implication holds, the second sentence is false in the same structure in which the first one is true.
Roughly, the first (correct) formalization becomes more easily true because the implication can be satisfied by falsifying the antecedent with a person not helping everyone who helps them, whereas the same is not possible for the second formula.