Use the expression tool. Make non printing expressions. On the playback tab, choose controller, then the midi number you need to edit, then enter a value. "0" is off, "127" is on. (or any value in-between).

That's the easy part! The hard part is "What goes in the controller slots?"  I'm trying various things that seem logical, and none of them are working.

Here's a screen of the number of variables I'm working with. If this were an actual physical switch, it would be easy: Hauptwerk would listen to the input, say "Oh that one" and be happy. But virtually, it doesn't even know what to listen for,

Hey, this worked great for the thumb pistons, and I didn't even have anything set in Hauptwerk. And since I've got 5 pistons on each manual to set, it gets me through what I need to do right this moment:

Page 128 in the Hauptwerk manual gives an overview of MIDI Learn and how it relates. I’d start there.

I find it easier to let Finale trigger MIDI in my DAW (Digital Performer). After recording, I then do all my editing in DP. 

I was hoping to avoid being incompetent in Logic Pro until I got the organ stuff done...recording the organ in HW, then overlaying a vocal in Logic. After an evening of pawing through both manuals, I think I'll just record while operating the master generals with a mouse. I can't even manage a channel change from 2 to 3, which the Finale manual gives instructions for.

Summary of what works, for the next guy:
Stop management: MIDI Tool/Continuous Data/Set to/program change [n], where n is the number of your master general in Hauptwerk. You can either attach this to a beat with the MIDI tool or to an Expression (dynamics are good). Make sure that you do this to the first beat, or Finale might give you something unwanted in HW. On the HW side, the master generals have to be set as "Momentary piston: MIDI program change", MIDI In = "Any available port",  MIDI channel = "Whatever channel that stave is set to in the Score Manager", "'On' program" = " same number as your master general/program change". You can control the thumb pistons this way too, but only for the division you're actually "playing " on; if you have a Great-to-Swell couple and you're playing the Great, you can work the Great piston but not the Swell. With the generals you can do all 3 (or more) divisions.

Channel change: For some reason it does not appear to be possible to send a channel change message from Finale to Hauptwerk. I have no idea why. The HW documentation talks about floating divisions without making them sound necessary.  I set them up, and saw how I could probably automate the switch that chooses between real divisions, but this also broke all my master generals. For the one piece where I HAD to have a channel change, I made a 'play' copy, added another grand staff, put all my Great material on one and Swell on the other, and gave them Channels 2 and 3 respectively, For most other situations, if you play the Great, you can use Great-to-Swell alone to play the Swell. (This is the St Annes Mosely I'm talking about; if your organ has Swell-to-Great, you can go the other way.). 

I was also told of a nifty tool:
"I don't have experience myself with Finale, but in case it helps you could potentially see what MIDI messages Finale is actually sending to Hauptwerk, and what virtual Hauptwerk controls (virtual keyboards, etc.) those messages are triggering, by temporarily turning on the 'Log all MIDI messages ...' option on the 'General settings | General preferences | Advanced ...' screen tab, playing the piece from Finale (or, probably better for ease of testing, a minimal test piece) then looking in the activity log ('Help | View activity log') in Hauptwerk.

There will be a lot of entries logged, so the less events your Finale test piece contains the easier it will be see the wood for the trees (so to speak). E.g. make a clean test Finale piece with just a single note (e.g. middle C) on a single stave, then a switch of MIDI channel for the stave, then just one more note on that stave. Likewise, avoid touching anything else in Hauptwerk while conducting the test, so as to minimise the amount of events that get logged.

You should also see the relevant virtual keys play (e.g. the two middle Cs) on the relevant virtual keyboards on the screen in Hauptwerk if it's working as you want it to."