I'm shocked that the red cross symbol is copyrighted. Don't tons of games use that symbol on healthpacks? I'm thinking specifically of military-themed games that have realistic looking medical equipment.
It is an officially protected symbol (along with the Red Crescent and, controversially[1], the Red Diamond), I'm not sure "Copyright" covers it, more than as a key piece of the Geneva Convention. It's been much degraded by 'casual' use[2][3] like game-graphics at least since the days of Doom and doubtless in various 8-bit applications in the decade or so beforehand (Civ used one or other of the rods, in its context of ddnoting medicine I can't remember off hand if that 'chose wrongly').
Filmmakers (and even "living history re-enactors") absolutely need permission to use Red-Crossed markings in their productions, so films like (or about) Dunkirk will get such permission, strictly vetted in the use to try to never defame the reputation of the organisation. This would make it tricky to depict "doctors gone wrong" or even "false-flag disguised as medics", even if it was depicting an actual such occurance for verimisilitude. There's probably ways to get around that (with agreement).
Video games have this issue, especially if "medic" class FPS characters are entirely supposed to be partisan and have at least a small-arm to go with their branded medikit, the use of each depending upon which team(s) are spotted out in the open with a reduced HP bar above their head. Maybe I would concede that in single-played runs through a Doomy dungeon, it's not any more inconceivable that someone from the in-game ICRCRC has ensured that official FAKs have been scattered as liberally upon pedestals, behind secret doors, etc (in whatever hell-dimension the player is currently fighting through) as that someone else has been similarly dropping ammo clips, body armour, etc.
For DF, you could probably get away with "green cross" symbol (IANALawyer!) if the snake-rod-of-whatever-form ever becomes too awkward to pixellate (say as shoulder-emblem for a character sprite where job-assigments/specialities are an overlaid infographic), but certainly the dorfs wouldn't really have any known cause to use the same symbology, it'd be for the benefit of the player (except where it isn't).
[1] Under the back-formed 'reason' that the cross is the colour-reversed Swiss flag (that coincidentally/by derivation only looks like a Christian cross) and the crescent is from the colour reversed Ottoman flag (without the star, and again only
coincidentally/by derivation is the islamic symbol) and thus neither is meant to be religiously-symbolic (but it doesn't harm if crescents and crosses are used in crescent/cross-tending cultures), the "Red Star Of David" suggestion was not well received. Apart from anything else, would the Hindu 'Red Cross' organisations ask for the Red Aum or even, if you can imagine it, the Red Swastik? Eventually they allowed the Red Diamond (red border, white fill) and that tends to be used in cross/crescent-'phobic' situations.
[2] Though it isn't used at all in the UK health service, not on its ambulances, etc... First Aid packs
may have it, but 'a cross' in white or embossed upon
green seem to be
de rigour, pharmacies employ a particular form of 'green cross', St. John's Ambulance have their own 'cross' symbol, and only actual British Red Cross resources (like their emergency vehicles) use, obviously legitimately, the exact symbology.
[3] Because of objections by the international/local Red Cross, unaffiliated mountain-medics in US ski resorts had to stop using red-crossed penants to indicate their nature, so they
reversed the Red Cross symbology... To then be theoretically using the Swiss Flag, again, double-inverted!