How to apply chain rule to a differential equation

deq = y''[x] + (epsilon - x^2) y[x];

deq /. {y -> (y[#^2] &)} /. x -> Sqrt[s]
(*(epsilon - s) y[s] + 4 s y''[s] + 2 y'[s]*)

Another option is to use the package MoreCalculus by Kuba

<< MoreCalculus`
diff[x_] := y''[x] + (epsilon - x^2) y[x] == 0;
DChange[diff[x], {x^2 == s}, {x}, {s}, y[x]]

$$ \epsilon y(s)+4 s y''(s)+2 y'(s)=s y(s) $$