You can define an overloaded constructor the usual way, but to invoke it you have to use the "new" keyword.
scala> case class A(i: Int) { def this(s: String) = this(s.toInt) }
defined class A
scala> A(1)
res0: A = A(1)
scala> A("2")
<console>:8: error: type mismatch;
found : java.lang.String("2")
required: Int
A("2")
^
scala> new A("2")
res2: A = A(2)