I can find no indication that Beck served during the war. At the time of the war, this very colorful Missourian appears to have been keeping himself quite busy criss-crossing the West, with one biographical profile stating that "he found time to cross the Western plains thirty-four times, before the days of the railroads." Note that he graduated from Yale, at the age of 18, in 1857. The first railroad across the plains was completed in 1869.
After graduating from Yale Beck became a doctor, and then a lawyer. Between 1857 and 1875 he handled legal cases in as varied places as Washington D.C. and California. During that time "His camp fires frightened away the wolf from what are now the sites of Omaha and Denver, and the mysterious country of San Juan echoed to the sound of his rifle long before silver and gold were even suspected in that country. In that vast territory on the northern border of Texas, marked on the map as an unknown region, over which roam herds of wild horses and buffalo, Mr. Beck delighted to pitch his tent."
Beck expressly declined to provide any information to one of his would-be biographers, the explanation of which might account for the dearth of present-day knowledge of this prominent person. "...kindly leave me to the obscurity I really covet...," he responded.
Beck's association with Confederate Colonel Alonzo Slayback after the war provides us no indication of Beck's wartime sympathies--Slayback's post-war law partner was Colonel James O. Broadhead, who was as staunch a Unionist as Missouri ever produced. Beck appears to have had a slave enlist in the U.S. Army, but that is not indicative of the politics of that man's owner.
James Preston Beck does appear in the Missouri Union provost marshal papers, having made a sworn statement on April 25, 1863 regarding monies he alleged to belong to the estate of his uncle, Preston Beck Jr.
James Preston Beck was the nephew of Preston Beck Jr., who was an equally colorful adventurer of the American frontier. He died in 1858 as a result of wounds received in a knife fight in Santa Fe, NM. He was a trader who was based out of Boonville, Mo.
There was a James Beck who served in Company I, 46th EMM. Beck was a resident of Howard County, and the 46th EMM was a Howard County unit. The Missouri State Archives card shows a middle initial of H. for this individual. However, given possible transcription problems, this disparity is not dispositive of the issue without going to the original source information. You might check the original muster rolls to see if the James Beck is your man. They'll provide place of birth, age, etc. You can cross reference to see if it indicates dob ca. 1839/pob Indiana....
Your best bet for information on possible wartime service might be in discerning a date of death for him, and then tracking down an obituary.
Good luck.
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My sources for the above information include--
"Saint Louis: The Future Great City of the World," (St. Louis: Hobart & Co.) 1876, pp. 684-687 (Includes photograph)
"History of Howard and Chariton Counties," 1883, p. 280
"Turmoil in New Mexico 1846-1868," By William Aloysius Keleher, 1952 (2007 reprint Sunstone Press), p. 128
"Officers and Students of Yale College 1856-57," New Haven: Ezekiel Hayes, 1856, p. 13
"Catalogue of the Delta Kappa Epsilon Fraternity," New York: Council Publishing Company, 1900, p. 6
"Reports from the Court of Claims Submitted to the House of Representatives," James Preston Beck, Administrator of Preston Beck, Jr., vs. The United States, 37th Congress, 2nd Session, Report C.C. No. 282, December 4, 1861, pp. 1-54
Missouri Union Provost Marshal Papers 1861-1866, reel F1277