Usually a Jenkins build job injects cookies environment variables to processes it starts ("JENKINS_SERVER_COOKIE" and "HUDSON_SERVER_COOKIE"). Which are inherited into all child processes. And after job is finished, those processes are killed if still are left running.
This does not work with TestComplete tests. My guess is, that this is because the TestComplete windows service is already running when job starts.
Somehow TestComplete/TestExecute and all processes which are started by the scripts (e.g.via Win32API.WinExec) needs these cookie injected. I hope the Jenkins plugin API provides the means to retrieve the current cookie value.