I've tried everything to get this game to run on my Windows 10 computer. I don't know what else to try. Can any of you guys help?

What problem exactly you have?

Game as most others not require any special tweaks even on Win10.
Reinstall video driver to one of the latest stable build. Disable in driver Control Panel any resolution\refreshrate increasing features. Enable Vsync (not any like Gsync). Set "Keep aspect ratio" for ratio option (if you have Intel integrated video you can do this in Intel driver and that work for both).
Set resolution in game properly, that supported by Display. Same for refreshrate - differ resolution = differ refreshrate. For most resolutions it 60 Hz. Some display support higher but only for maximus\native resolution... or vise versa higher supported only in 800x600.
I dont remember if that option editable in ingame settings or in game ini files or nowhere. Check it manually.
Game support maximum 1280x960 and to set this you must have FullHD display. 1280x1024 is a 5:4 aspect ration and not properly scaled on 16:9 widescreen even if it FHD. Of course lower wide display's cant handle any of this.

Here the others general solutions. Install latest DirectX 9 (June 2010?) and download DirectX web installer to check updates. DirectX 9 never be pre-installed in any Windows.
Check Windows updtaes. For Win10 make sure you have one of the latest stable public builds.
Enable Legacy DirectPlay in Legacy components in Enable\Disable Windows features in Win Control Panel. Completely disable Windows DEP security features -google it or check GOG\Steam\Microsoft support sites. To disable completely you must start and use SySWoW64\cmd.exe as administrator

Did you use GOG version? Which one - offline or Galaxy? or Steam or Retail?

Temporary disable antivirus soft if you use any other than Windows Defender.
For Galaxy:
Reinstall Galaxy to C:\Games\GOG\Galaxy\ path as admin. Not any special symbols or Cyrillic letters. No spaces. Set on Galaxy exe "adminrun". Start it and check for client updates.
Then reinstall game from reinstalled Galaxy to path (set it in general galaxy settings) similar to above. Like that:
X:\Games\GalaxyGames\Army-Men-RTS\ or simple X:\Games\Army-Men-RTS\
if you start game from exe file - set "adminrun" on it. If you use Galsxy to start game make sure that galaxy exe still started as "from administrator" - it apply to all further exe's (all game).

Verify game files in Galaxy client after downloading. twice. Go to game folder and remove any other compatibility even if that enabled by Galaxy client.

Check game ini files.

For offline build:
Did exe file have digital signature? Reinstall it as "from admin" to path X:\Games\Army-Men-RTS\ and better if that not on C:\ drive. Make same things as for galaxy version - enable adminrun and disable other comp-ty tweaks.

If that not help - enable "Disable fullscreen optimisation". And if that not help - some other tweaks in Compatibility. But as i said above game not require any of this. Only if some other broken Windows10 "security" updates.

Here the further must-read links:
https://support.gog.com/hc/en-us/articles/212454069-Nothing-happens-after-double-click?product=gog
https://support.gog.com/hc/en-us/articles/115002094065-Army-Men-RTS?product=gog

(unofficial Wiki and of course not 100% accurate!)
[url=http://www.pcgamingwiki.com/wiki/Army_Men:_RTS]http://www.pcgamingwiki.com/wiki/Army_Men:_RTS[/url]
http://www.pcgamingwiki.com/wiki/Microsoft_Windows
http://www.pcgamingwiki.com/wiki/Troubleshooting_guide
Other GOG\Steam\Microsoft\google links.

Check it, if this help for you (optional)?
https://support.gog.com/hc/en-us/articles/115003260089-Disabling-Windows-10-Game-DVR-?product=gog

If nothing help - reinstall Windows Vista or WinXP. Or forget about PC gaming.