Yes, your app built and compiled on WIndows should work on your Linux or Solaris machine.
If you used Java 6 u 21, make sure the JRE on the Unix machine is the same version.
Make sure that the ArcGIS Runtime on the Unix machine is properly licensed and the init scripts are executing successfully so that the jars link to the arcobjects runtime libs properly.
What is your target Unix platform? Linux? Solaris? What versions of these operating systems are you trying to run on?
Thanks!