I was able to find a solution to this problem, but only by using a couple extra tools that are later in the book$^{1}$. So far the book only covered basic divisibility, $gcd$, and the fundamental theorem of arithmetic; it did not cover modular arithmetic, and altough we did cover the division algorithm, we did not cover divisibility rules (i.e. if a number ends in $5$ or $0$, then it is divisible by $5$). Is there any way of proving this with only the above tools? (I will point out what I used from future chapters in my solution)
My Solution
Suppose $10 \nmid p^2-1 = (p+1)(p-1)$. Then $5 \nmid (p+1)$ and $5 \nmid (p-1)$.
Odd primes can only have last digits of $1, 3, 7, 9$ (I used the divisibility rule that a number ending in $0$ or $5$ is divisible by $5$, which is in the next chapter). Since $5 \nmid (p+1)$ and $5 \nmid (p-1)$, the last digit of $p$ is either $3$ or $7$. If we write $p$ as $10n+3$ or $10n+7$, then square and add $1$, we get a multiple of $10$. (The fact that any integer with a last digit of $k$ can be written as $10n+k$ is also something from a future chapter)
Elemntary Nuber Theory by David Burton 6th ed., Section 3.1 # 10
Answer
If $p\neq 5$ is an odd prime, its square $p^2$ is also odd, thus $p^2-1$ and $p^2+1$ are both even.
Now, since an odd prime $p\neq 5$ must (as you mention in your post) be: $$p\equiv1,3,7 \textrm{ or }9 \mod 10$$ its square will be
$$
p^2\equiv1,-1,-1\textrm{ or }1 \mod 10
$$
which answers your question.
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