##Scala, 222

object O extends App{def q(i: Int)={val x=math.sqrt(i).toInt;x*x==i}
println((args(0).toInt to args(1).toInt).maxBy(n=>{val s=n+""
s.tails.flatMap(_.inits).filter(x=>x.size>0&&x(0)!='0'&&q(x.toInt)).size.toFloat/s.size}))}

(Scala 2.9 required.)

Scala, 222

object O extends App{def q(i: Int)={val x=math.sqrt(i).toInt;x*x==i}
println((args(0).toInt to args(1).toInt).maxBy(n=>{val s=n+""
s.tails.flatMap(_.inits).filter(x=>x.size>0&&x(0)!='0'&&q(x.toInt)).size.toFloat/s.size}))}

(Scala 2.9 required.)