"The quantity x minus y, all squared."
Commenters have suggested using "quantity" to indicate that x-y is a single expression. I suggest saying "all squared" as a reminder of that.
(Source: mathematician and educator who says/hears this often.)
Edit: I'll add some more thoughts to make this answer more complete and simultaneously address a few comments.
Ultimately, there is not one conventionally "correct" way of reading such an expression out loud or in one's own mind. The goal here is to convey the concept of taking two numbers, which are called x and y, finding the difference x minus y, and then squaring that result. Indeed, mathematical notation is created to fill that purpose! We use the expression (x-y)^2 to mean exactly that idea I just described in words, and this expression will always be unambiguously interpreted in that way.
So, when someone asks, "How do I read that expression out loud?" or "How do I say that expression in my head when I'm reading it in a book?", they are asking for a way to translate that expression from mathematical language into natural language, and this may introduce ambiguities. In some sense, the "correct" way to write/read this expression is the one that is in mathematical language: (x-y)^2 There is not one officially correct way to translate this into natural language. Rather, the best thing to do is to say/write anything that will be interpreted unambiguously as that expression. I suggested "the quantity x minus y, all squared" because I am confident that 100% of my colleagues and students would read those words in natural language and translate them into (x-y)^2 in mathematical language.
Someone suggested "the quantity of x minus y" instead, and this is not really more or less correct in any way. It is just different. I am also confident that my colleagues and students would interpret that as (x-y), but I also suspect that some of them (like me) would bristle a little at the "of", not because it's wrong but because it just sounds redundant and unnecessary. Although, if I were keeping track of how I say out loud what I suggested, I would probably go, "The quantity [short pause] x minus y [short pause] all squared". I suspect the suggested "of" is analogous to the brief pause I would use when saying this out loud.
But ultimately, I hope this sheds more light on these ideas and why I think my answer is best.