Nearly 10 months and no answers, disappointing to say the least.

Right now I'm waiting on "GOG Support" to respond to this very question, but have found the following, both of which say Destroy All Humans will run on Windows 7 (64-bit) and Windows 8 (64-bit) (plus W-10 64-bit of course). That said, I have not been able to confirm these claims from GOG or any other sources. If or when GOG Support gets back to me, I'll try to find my way back here and let you and everyone else know what they said. Right now this game is on sale but I don't know that I'll have the answer from GOG before the sale runs out on 6/26/2021 (they say they are way behind)?

Under "System Requirements" at the following link (this link was supplied by an automated GOG email "like response" to my question about this subject) it says:
"Windows 7 and 8.1 are supported, but some AMD graphics cards might show weaker performance on those OS versions."
Link: [url=]https://www.pcgamingwiki.com/wiki/Destroy_All_Humans![/url]

At the following link (a source I found searching for answers) they say it will run on Windows 7, 8, and 10 (but don't specify W-8 and/or W-8.1), but all have to be 64-bit operating systems (which the other link also says). Note: They call this version "Destroy All Humans Remake", but the release date appears to match the GOG version.
Link: [url=]https://www.kitguru.net/gaming/joao-silva/destroy-all-humans-remake-system-requirements-and-release-date-revealed/[/url]

For what it's worth, the Destroy All Humans DEMO (Free>) from GOG ran perfectly on my Windows 7 Professional (64-bit) system which has an older Intel Sandy Bridge-E Core i7-3820 (Quad-core with 8-Threads), an NVIDIA GTX-980 (4GB VRAM), with 8 Gig's of system RAM (4 X 2GB in Quad Channel mode), if it helps anyone?

I got a surprise quick reply from GOG this morning.
I can't believe I completely missed it but, on the GOG page for this game, at the bottom of the requirements section, is the following statement under "Other":
"Windows 7 and 8.1 are supported but some AMD Graphics Cards might show weaker performance on those OS versions."

This matches up with one of the links I posted above, so in some or many cases, this game will run on Windows 7 and 8.1, if the machine is fairly strong. AMD video card users may have performance issues requiring the need to turn down graphic details for smooth performance.

Edit:
I left out a detail in the response GOG Support gave me, which is:
" You might need to lower some graphical settings to not go into 4GB VRAM territories"

I take this to mean that my GTX-980 with 4GB of VRAM is "Marginal" to run the game with graphic details turned up to high levels.

This all brings me to make a suggestion for anyone who's not sure if their PC can run the game. As I did months ago, I suggest that you download the Free Demo version of the game, and see if it runs on your system. If it does run, the chances are very good the full game will also run.

In the event it helps anyone, here are the full spec's on the machine I used for the demo (I'm going to buy the full game, but just haven't done so yet):

Processor: Sandy Bridge-E Intel Core i7-3820 @ 3.60GHz, 3.80 Max Turbo,
Quad-Core, w/Hyper Threading (8 Threads), Quad-Channel Memory Controller,
PCI-Express 2.0 with 40 Lanes, LGA-2011 Socket (2nd generation i7).

Motherboard: ASUS (Republic Of Gamers - ROG) Rampage IV Formula.

RAM: 8GB DDR3 (4 X 2GB In Quad Channel Mode), Supported: 1066/1333/1600.

Video: (1) NVIDIA GeForce GTX-980, 4GB GDDR5 VRAM (Installed in early 2019).
Processor: Maxwell GM204, 256-bit Interface, Bus: PCI Express 3.0.
2048 Cuda Cores, HDMI 2.0 support, Outputs: DVI-I, HDMI, 3 Display-Ports 1.2

Sound: Creative Sound BlasterX, AE-5 Plus, PCI-Express

Good luck!