When someone says two things are equivalent, what that means is that they give the same result for some test; they are equivalent with respect to that test. For instance, in Texas Hold 'Em, getting dealt the ace of spades, then the ace of diamonds, is equivalent to getting dealt the ace of diamonds and then the ace of spades, because strength of hands doesn't depend on the order that you received your cards. So to answer the question "How are these functions equivalent?", we have to know what test they're equivalent with respect to.
A utility function is a tool that's used to determine, given two options, which one someone would take. The idea is that we look at what utility each option has, and the person will take the one with the higher utility. For this test, we only look at which is larger, not the absolute size. So if you have two functions $f$ and $g$ that always give the same order, they are equivalent with respect to this test. That is, given any two options $a$ and $b$, if $f(a)<f(b)$, then $g(a)<g(b)$.
Now, if we set $f$ to $(x,y) \rightarrow xy$ and $g$ to $(x,y) \rightarrow (xy)^2$, we can introduce $h: u \rightarrow u^2$. Then $g(x,y) = h(f(x,y)$. In other words, $g$ is $h$ composed with $f$. If we assume that $x$ and $y$ are positive, then $h$ is monotonically increasing, and thus composing it with any function doesn't change ordering; if $n<m$, then $n^2<m^2$ (again, for positive numbers). Any time two functions are such that one can be written as a monotonically increasing function composed with the other, the two functions are equivalent with respect to ordering.
However, it is important to note that utility functions are also used for evaluating a concept called "lotteries". This is where someone chooses between options that involve different probabilities of different outcomes. For instance, someone might have a choice between "1% chance of getting \$1m and 99% of getting nothing" versus "100% chance of getting \$100k". When evaluating lotteries, the functions that you list are not equivalent when applied to the outcomes.
