How can I compile and run a Java class in a different directory?

I might be misunderstanding the question, but you can compile with

javac /home/MyJavaFile.java

This will create MyJavaFile.class in /home

You can then run it by including /home on the classpath. e.g.

java -cp /home MyJavaFile

If you want to generate the class file in a different directory then you can use the -d option to javac.