what is the eligibility to receive this?
Department of the Army, Navy and Air Force.
Medal of Honor
There are three versions of the medal, one each for the Department of the Army (United States Army), Department of the Navy (United States Navy and United States Marine Corps) and Department of the Air Force (United States Air Force). 
what is the 4th missouri infantry regiment? An infantry regiment that served in the Confederate States Army. when was it formed? On April 28, 1862. which war is associated with it? American Civil War. what was the cause of the war? The enslavement of black people. between whom did it take place? Northern states and southern states. what was the role of women in it? Between 400 and 750 women had served as soldiers. can you name a few of them? Mary Edwards Walker. can you tell me something about her? The only woman to ever receive the Medal of Honor. who presents the award? The President of the United States.
wiki:25371909
Medal of Honor [SEP] Introduction
U.S. awards, including the Medal of Honor, do not confer post-nominal titles, but the official abbreviation (often informally appended after the recipient's name) is "MOH". There are three versions of the medal, one each for the Department of the Army (United States Army), Department of the Navy (United States Navy and United States Marine Corps) and Department of the Air Force (United States Air Force). Despite not normally being a part of the Department of the Navy, the United States Coast Guard awards the DoN's variant of the medal. The Medal of Honor was introduced for the Department of the Navy in 1861, soon followed by the Department of the Army's version in 1862.