In the latter half of the 19th century major European militaries and the US military had "medical corps" enlisted men of one stripe or another who provided first aid on the field, assisted with medical clerical/supply duties, pharmacy dispensing, and bedside care in post hospitals. This was a more formalized system than the earlier system where there were limited numbers of "medical enlisted" and much of the transportation of the wounded and "nursing" care was done by line soldiers (and/or bandsmen) detailed to this duty - and almost universally accounts state that the line soldiers detailed for medical work were usually wither consider the "losers" by their company officers or those who were physically unfit for line service (recovering wounded for example).
Having said that, one reason men were used for bedside care in most militaries in peacetime (and initially in war) was a reluctance to have soldiers "coddled" by women as well as keeping women away from the ugliness of military life even in peacetime. Formal and permanent female nursing establishments in militaries did not arrive until the end of the 19th/beginning of the 20th centuries.
A BIG roadblock against male nurses becoming "common" in the mid-19th century is the fact that "medical care" was seen as a very female occupation. Even though women only began to be allowed in to medical schools in the middle third of the 19th century, physicians (especially in the military) were seen as operating in a "feminine" sphere since most health care was provided in the home under the care of women (wife/mother/sister). Lots of other reasons why nursing and men don't belong in the same sentence in the time of Florence Nightingale. Lots of references in the medical history literature discuss this issue.