Is this somehow integrated with Java or does it work with a programming language of my choice?

Specifications aren't tied to any specific programming language - you're outlining how the algorithm should work, not the implementation.

(You can write Java code and invoke it during model-checking if you really need to, but I haven't come across any situation where it's actually necessary.)

Ah, okay.

I had the impression it was about somehow writing a program in a more strict language and then exporting it to Java.

I don't do much algorithmic design.

I've written specifications of Ruby and Python codebases before, and worked with people who wrote Java, C, and Haskell code. It's pretty versatile!