Fold3 has this at the following link http://www.fold3.com/page/812_the_civil_war_in_bollinger_county/:

"Marble Hill lies along the Old Military Road from Jackson to Greenville, a road much travelled by both Union and Confederate troops during the war. It is along the Old Military Road that visitors will find the grave of the lone Union soldier. During the war a group of Union soldiers travelling the road stopped at a home along the way to ask for milk for a wounded soldier being carried in a wagon. The injured man died a short time later and was buried beside the road. For many years the grave had no marker. One evening a couple passing by the grave noticed something white. They discovered a tombstone with the inscription 'W. Woods, Union Soldier. Died For His Country.' No one ever learned who had placed the marker."

Note that this lists the name as "Woods" rather than Wood (I wonder which was on the original marker...or if it was even correct.) Not much to go on as it doesn't give any indication of timeframe, and therefore makes it difficult to narrow a list of candidate units. It doesn't help that W. Wood or Woods is a very common name for the period.

I did come across a tantalizing listing of a "CW burial" record of a Wilson Woods in the Missouri S.O.S. database, but it doesn't have an image of the card and doesn't list a unit or anything else. The one Wilson Woods given in the same sweep of the SOS appears to be one who enlisted in the Dallas Co. Home Guard and later the 14th MSM before deserting in Buffalo (Dallas County.) There appears to be a duplicate of this desertion record for Wilson Wood (no "s") for the 8th MSM under Capt. Burch. This has to do with the consolidation of the MSM regiments as I understand it.