I found a handmade pillow in an antique shop and it is hand embroidered with the words WW1 veteran (it had some other details on it as well.) It definitely looks authentic for the time period. I researched a little and found that embroidery was taught and used by disabled soldiers in the hospitals as a form of therapy for British , New Zealand and Australian soldiers. Was this practice also used in the United States?
I'm not aware of any record of U.S. soldiers using embroidery as a form of rehabilitative therapy, but it was widely used among Canadian soldiers at British hospitals, so it's also possible that your piece was originally made by a Canadian solider, or by a U.S. soldier whom was treated at a British or Canadian rehab hospital.
I also want to point out the forgotten practice of the embroidered post-card, which as a cultural practice that pre-dates the First World War, especially in France.