$$ \forall x \exists y ((x \lt y) \implies (x^2 \lt y^2)) \space where \space x,y \in \mathbb{R}$$
I used proof by contradiction and found that it is true. However my professor thinks it is false because if for example x=-5 and y=-2 then $x^2 \lt y^2$ would be false.
To my understanding, the statement means that for every x there exist at least 1 y that satisfies the statement. Am I misunderstanding this or is my professor incorrect?
Unfortunately, it does appear that your professor is incorrect. If the statement was for all x AND for all y, where x is less than y, then what your professor said would make sense. But in this case you're correct, it's simply talking about the existence of at least one y for each x that satisfies the statement.