Whether or not you decide to fast is a personal choice. HOWEVER, allow me to say that you must consider how the fast will affect your performance the day after your fast regardless of your having broken the fast 24 hours later. With the rigor of basic training and the related ramp up of your metabolism, your body will literally be starving for nutrition and you will feel it. Ask yourself, how often do you now perform those taks that will be imposed on you at basic training on a daily basis: never or rarely at most. The day after a fast is a day of phyiscla recovery for anyone, let alone a Soldier on a rigorous training schedule. The biggest concern is for your safety. Since you don’t know the limits of your body based on participation in basic training activities yet, I (as a nurse) and many Rabbis would discourage you from participating in the fast. Imagine yourself at the top of your repelling tower the day after your fast and potentially feeling dizzy. Or on the firing range with a lodaded weapon and passing out. At every High Holiday, we always provided our Soldiers with some amount of nutrition during the fast for their own safety. Consider this, please - choose life.
The answer to your original question is “no.” You will not be permitted a day off to fast, but rather you will be permitted a day off to attend the Yom Kippur service (essentially starting from the evening before to the breaking of the fast meal). Please ensure proper coordinatioin through your chain of command and the Jewish Lay Leader or Chaplain.