Clearly, $3^2=9$ is a perfect square, all of whose digits are $7$, $8$, or $9$. Are there any other perfect squares with this property?
This is an interesting question that does not seem to be solved yet, coming from AoPS (https://artofproblemsolving.com/community/c6h1928519). As duck_master seems to show, it should be impossible to solve this problem by analyzing the quadratic residues modulo $10^n$ for some $n$.
I strongly suspect the answer is no. I have been running a Python script for quite some time, and it has checked squares up to $(50,000,000,000)^2$ with no results (unless I messed up the code).