I have not had luck with induction here. However, here are two very unusual observations I have made:

for n even and for n odd. In other words, the decimal expansion seems to be tending towards an alternating pattern between .375 and .875. To be a perfect square, this decimal would of course need to be exactly zero, so proving this observation would imply your hypothesis.

Since it does alternate, you may try using induction where

And don't forget the identity