This caught my eye due to the shield design and the branch letter looking a lot like yours. So looking at the Mexican American war period led to several buttons being found with the same head, arrow, olive branch configuration:

See http://www.civilwarbuttons.com/uscentrlist.htm and run down to uc91 and below.

According to this source published 1895, the letter insignia in the shield went away as of 1851. It also states that the branch insignia on the buttons were post 1812. http://www.qmfound.com/changes_in_the_army_uniform_1895.htm

I'm not a button expert by any means but Missouri sent a good number of men to the Mexican American war including several companies of Infantry. A number of those men ended up fighting in the Civil War in Missouri on both sides.

So I'm guessing 1840's Union Infantry Button. Perhaps made specifically for the Mexican American war. see http://www.history.army.mil/html/museums/uniforms/survey_uwa.pdf for samples that look real close to yours except for reversed olive branch and arrows and then consider was it easier to reverse hands than to turn the head in order to signify war time button?

FWIW
John R