EDIT: Apparently you can, but the rules are different than for a regular job. You can only get unemployment for 6 weeks. AND this should have been covered in the class you would have taken about re-entering the civilian world.
Source: Asked my boyfriend, who was in the Navy.
Depends on your state's laws, but I'm pretty sure the answer is absolutely not.
See, if you were honorably discharged, then you were discharged for one of three reasons: 1) you finished your term of service and chose not to re-enlist; 2) you had an injury or other medical condition that prevented you from doing your military duty; or 3) you retired from the military.
Since you are trying to become a police officer, I am going to assume No. 2 does not apply. I am also going to assume No. 3 doesn't apply, because if you retired from the military, you'd be drawing a pension and wouldn't need unemployment.
So that leaves us with No. 1 - you chose not to re-enlist. Which, in layman's terms, means you "quit" your "job". Since you would have had a job in the military had you re-enlisted, but you chose not to, it was therefore YOUR choice that left you unemployed, not the military's choice.
Therefore, you don't get unemployment.
But again, your state's laws may differ. You can always check and see what they are in your state.