Prove that every CFL has at least one infinite equivalence class
The language of palindromes over $\{0,1\}$ is context-free, but every Myhill–Nerode class is a singleton. So your claim is wrong for context-free languages (unless the alphabet is unary). In contrast, regular languages have a finite number of Myhill–Nerode classes, so at least one of them has to be infinite. The same is true for context-free languages over a unary alphabet, since all of them are regular.