Thanks to this post: on Steam, I was able to get this to work. Using DXWND (which I haven't used in a while) worked great. After running DXWND for the first time, it'll create a dxwnd.ini file. You can copy/paste the following into it (change the game location to wherever it is installed on your system):

[window]
posx=665
posy=215
sizx=588
sizy=372
[target]
title0=Haegemonia Legions of Iron
path0=H:\Games\GOG Games\Haegemonia - Legions of Iron\HGM.exe
launchpath0=
module0=
opengllib0=
ver0=0
coord0=0
flag0=806355140
flagg0=134217744
flagh0=1040
flagi0=2228224
tflag0=0
initx0=0
inity0=0
minx0=0
miny0=0
maxx0=0
maxy0=0
posx0=0
posy0=0
sizx0=1920
sizy0=1080
maxfps0=0
initts0=0
winver0=8
maxres0=0

notes0=
registry0=
flagj0=0
flagk0=16384
swapeffect0=0

The game is 4:3, so it will stretch to the resolution you specify in the sizx0= & sizy0= section. Also, you can force the game to use whatever resolution by going into your Registry, which I followed from the good folks over at WSGF.org. I hope this helps others.

Works like a charm on the original game, but I'm having some difficulties with The Solon Heritage.

The games refuses to save it's settings, every time I start the game they are back to default, causing the window borders to be visible and gives me a double mouse cursor if I move it outside the window border. As soon as I set the options to what I want them to be it's full screen once more, but it is annoying having to do this every time.
Changing the options in DXWND to run the game in full screen means it stays that way, but still the options reset.

Digging around in the registry seems to indicate The Solon Heritage has very few settings of it's own, but seems to share most of them with the original game. This is supported by the fact that the first time you run the original after the expansion it too will show up in a window, but restarting it will cause it to restore the settings from the registry.

The settings in the registry never change, but still The Solon Heritage refuses to load them, instead going with it's own built in default.

The settings I'm trying to change:

Resolution from 1024x768x24 to 1280x960x24
Texture filtering from Normal to Anisotropic
Autosave frequency from 10 min to 5 min

Had no problem changing them in the original game, but The Solon Heritage seems to have a mind of it's own.

No idea if this problem affects anyone not using DXWND as well.

I seem to have found a solution.

Unchecking "Run in window" on the "Main" tab, removed the problem with a visible border and unchecking "Emulate Registry" on the "Compat." tab allows the game to save it's settings. The Solon Heritage now have it's own settings saved in the registry alongside the original game. :)