For the reference of anybody looking to play the game on Windows 7, I have been running Army Men with no problems for some time now.
A few things to do to increase the likelihood of these games working:
-Install on a standard HDD, not a SSD.
(The game won't get any benefit from the SSD anyways, and it could cause compatibility issues.)
-Install in a short path on the same drive as the operating system.
(Unless the OS is on a SSD, then just install to a short path.)
(C:\Games\ArmyMen for example.)
-Play around with the compatibility settings tab for a bit.
(For me, running it in Win 98 compatibility caused crashes, while Win XP SP3 worked perfectly. On my last computer, Win 98 mode worked well, but in Win XP mode it didn't even launch. Again, play with the settings until you find something that is stable for you.)
-Run it as an administrator.
(In the compatibility tab, make sure to run it as an administrator.)
-White list the directory in your antivirus.
(Antivirus programs, especially ones like Norton or AVG, are extremely over aggressive in their active scanning, and some older programs will crash if their process is interrupted by an antivirus program.)