Here are the presidents who did not serve in the military.
11 or 12 -see LBJ note - of the total of 43 chief execs:

John Adams
John Quincy Adams
Martin Van Buren
Grover Cleveland*        
William H. Taft        
Woodrow Wilson        
Warren G. Harding
Calvin Coolidge
Herbert Hoover        
Franklin D. Roosevelt
Bill Clinton        

*He was drafted during the Civil War, but paid $150 for a substitute (a legal option under the terms of the Conscription Act of 1863, and his substitute survived the war).   {Alltime chickenshit president award winner.}

{Lyndon B. Johnson was in the US Senate when his joined the Reserves. He was credited with one B1 bombing mission as "an observer", but there is strong evidence that he was never on the plane. Read fake, as with Tonkin Incident. Yep, classic LBJ.}