Baquet was born in Paterson, New Jersey, one of eight children born to Camille Baquet (or Baquett), a law professor and translator from Paris, and Harriet Stuart Lord, the daughter of English immigrants. He grew up in Burlington, New Jersey. He was mustered in as a private in Company I, 16th Pennsylvania Volunteer Cavalry on September 13, 1862. He served with the Pennsylvanians until April 1, 1863, when he was commissioned as a second lieutenant in Company A, 1st New Jersey Volunteer Infantry, filling an officer vacancy within the Company that had existed for over a month. He then served until June 23, 1864, when his enlistment expired by law, and he was honorably mustered out of Federal service. During his 14-month tenure with the 1st New Jersey, his regiment saw combat at the Battles of Chancellorsville, Gettysburg, the Wilderness, Spotsylvania, and Cold Harbor.