It is my understanding that it has not yet been determined if it is possible to construct a $3$x$3$ magic square where all the entries are squares of integers. Is this correct? Has any published work been done on this problem?
The existence or not of a non-trivial integer 3x3 magic square of squares is STILL a unsolved problem.
The quoted reference to Kevin Brown's web pages only discusses an extremely special configuration of numbers, which does not exist. The page does NOT claim to prove non-existence for all possible magic squares.
If you are interested in this topic you should consult the web-site
which gives lots of details and references.
There is a claimed proof of nonexistence by J. C. Ferreira, but is it correct? https://arxiv.org/abs/1506.06621