Contrapositive of the statement "If two numbers are not equal, then their squares are not equal" is:

We begin by recalling the basic logical rule for conditional statements. For any implication of the form $$p \rightarrow q,$$ the contrapositive is obtained by first negating the conclusion $$q$$ and then making this negation the hypothesis, while simultaneously negating the original hypothesis $$p$$ and making this negation the new conclusion. Symbolically, the rule is stated as:

$$\text{If } p \rightarrow q,$$ then its contrapositive is $$\; \lnot q \rightarrow \lnot p.$$

Now we identify the parts of the given English sentence. The original statement is:

“If two numbers are not equal, then their squares are not equal.”

We translate this into symbolic form. Let

$$p : \text{“the two numbers are not equal”},$$ $$q :$$ “the squares of the two numbers are not equal” $$.$$

Thus the given statement is precisely $$p \rightarrow q.$$

To form the contrapositive, we negate $$q$$ and $$p$$ in turn:

First the negation of $$q$$ is

$$\lnot q :$$ “the squares of the two numbers are equal” $$.$$

Second the negation of $$p$$ is

$$\lnot p : \text{“the two numbers are equal”}.$$

Applying the rule $$\lnot q \rightarrow \lnot p,$$ we obtain the contrapositive sentence in everyday language:

“If the squares of two numbers are equal, then the numbers are equal.”

Now we compare this derived sentence with the options provided. Option C reads:

“If the squares of two numbers are equal, then the numbers are equal.”

This matches word for word with our contrapositive.

Hence, the correct answer is Option C.