liste = Table[Random[Integer, {1, 100}], {100}]

Select[Sqrt[liste], IntegerQ]^2

Flatten[Position[liste, x_ /; IntegerQ[(Sqrt[x])]]]

\(Flatten[Position[liste, _?\((IntegerQ[\@#] &)\)]]\)

\(Cases[liste, _?\((IntegerQ[\@#] &)\)]\)