In java 1.1, a partial solution was provided - all classes can have an instance initialiser, a block of code that runs when an instance is constructed - this works essentially like a default (i.e. noarg) constructor. This is better, but it still doesn't allow one to define constructors with arguments.
This page shows a little trick were one can simulate this functionality. The example below shows a non-anonymous baseclass which a non-default constructor that then calls a regular method. The anonymous subclass then overrides this method.
The example below shows an anonymous class that subclasses MouseAdapter that features a one-arg pseudoconstructor.