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I googled this question and saw this answer but I wasn't satisfied.

Squirrl
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    It depends on what kind of infinity you're referring to. – Shaun Jul 15 '14 at 09:28
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    How about cardinals? – John Smith Jul 15 '14 at 09:38
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    @JohnSmith There are no additive inverses with cardinals (indeed, cardinal addition is not generally cancellative), so "negative infinity" does not make sense. – anon Jul 15 '14 at 09:39
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    Your doubt is due to the "ambiguity" of infinity. If you are speaking of numbers (natural, integer, rational), $∞$ and $−∞$ are not numbers. If you are considering calculus, the usual way to write $lim_{n→+∞}n^2 = ∞$ is a simple way to express "in symbols" the fact that the sequence $s_n:=n^2$ diverges, i.e. that for any $M$ you can find $n$ such that $s_k = k^2 > M$, for all $k > n$. – Mauro ALLEGRANZA Jul 15 '14 at 10:03

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Lemma. Let $\{p_n\}_n$ be a real sequence such that $\lim_{n \to +\infty} p_n = -\infty$. Then $\lim_{n \to +\infty} p_n^2 = +\infty$.

In this sense, your conjecture is correct, and the previous lemma can be easily proved by using the mere definition of limit.

Of course, asking what the result of an algebraic operation on a symbol that is not a number is, may lead to different answers. First of all, you should ask (yourself) what is $-\infty$ and what is $+\infty$. If you are merely thinking of limits in calculus, then everything goes well. If you are dealing with infinity as a different object, you should expect some trouble from every partial extension of the usual arithmetic rules to it. But you should tell us more.

Siminore
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    Please explain the downvote. – Siminore Jul 15 '14 at 09:31
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    It was a little harsh of me, I admit. I'm sorry. There's a subtle issue here of what kind of infinity @Squirrl is referring to that this answer doesn't (didn't) address. For instance, in the extended complex plane, $-\infty$ "is" $\infty$. (You've now made that clear so, yeah, +1 from me.) – Shaun Jul 15 '14 at 09:37