Start a detached background process in PowerShell

Consider using the task scheduler for this. Define a task and set it without any triggers. That will allow you to simply "Run" (manually trigger) the task.

You can set up and/or trigger scheduled tasks using the ScheduledTasks powershell module, or you can use the GUI.

Use jobs for this:

Start-Job -ScriptBlock {
  & java -jar MyProgram.jar >console.out 2>console.err
}

Another option would be Start-Process:

Start-Process java -ArgumentList '-jar', 'MyProgram.jar' `
  -RedirectStandardOutput '.\console.out' -RedirectStandardError '.\console.err'