Interesting information. seems that the symbol is protected by treaty under the Geneva Conventions:
"In order to ensure universal respect for the emblems, the Geneva Conventions obliged their signatories to forbid any other use of the names and emblems in wartime and peacetime.
Nevertheless, the misuse of the emblem is widespread and it is often used as a general symbol to indicate first aid, medical supplies and civilian medical services especially walk-in clinics. Misuses appear in movies (A notable example is The Living Daylights, wherein narcotics were disguised as Red Cross care packages, as a plot device), on television, and in computer software and games. Service companies, such as those for car repair or lawn maintenance, tout themselves as service "doctors" and incorporate medical symbols to promote themselves.
In 2006, the Canadian Red Cross issued a press release asking video game makers to stop using the red cross in their games; it is an especially common sight to see first aid kits and other items which restore the player character's health marked with a red cross."
http://posterwire.com/dont-cross-the-red-cross/ but in this modern age of intellectual property litigation, companies have been removing the red cross logo to avoid possible infringement:
A red cross symbol is not a generic symbol for first-aid, emergency, hospitals, healthcare or medical services, products or personnel. The red cross symbol is a trademark owned by the American Red Cross and protected by federal and state trademark law, unfair competition law and anti-dilution law, and it is also protected by federal criminal law (See 18 U.S.C. 706, 917). The American Red Cross vigorously pursues those who infringe American Red Cross trademarks
https://www.scribd.com/document/3761982/Red-Cross-Emblem-Brochure A “misuse” is another word for “infringement,” andit describes instances in which an unauthorized partyis using the Red Cross emblem without permission fromthe American Red Cross. Often misuses occur when people incorrectly treat the Red Cross emblem as a generic symbol for first aid stations, ambulances,emergency, health care or medical products, services or personnel. A misuse can also occur when an authorizedparty uses the Red Cross emblem incorrectly.A misuse or infringement of the Red Cross emblem isany cross of equal or substantially equal vertical andhorizontal “legs” that are colored red or a shade of red.Misuses include red crosses that are:
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Slanted or italicized
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Thin or thick
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Outlined in a color other than red
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On a background other than white
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Included with a figure, symbol or wordsuperimposed on it
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Included as an element of a logo
Seems like the safest alternative is not to use a cross at all.