It's interesting how things are different in different parts of the country. Here in Illinois (and Southern WI and Northern IN), boxes can mean two things.
The first, and most common, is a "Box Card" that is tied to MABAS (Mutual Aid Box Alarm System). MABAS is an organization that provides aid to departments from other departments. If Dept A has a fire, Dept B, C, D, etc, will go with the understanding if Dept B has a fire, the others will go there too.
A typical fire works this way in IL/WI/IN. The initial report of a fire is a "Still Alarm/Full Still Alarm/Still & Box Alarm/Working Fire/Firebox" or some similar terminology. These are typically one or two department responses consisting of between 1 and 3 engines, 1 and 3 trucks, a commander or three, and maybe an ambulance or two. This is all done without activating the MABAS system. When that initial response is on scene and determines additional resources are needed, they request a MABAS alarm. MABAS alarms are the Box, 2nd, 3rd, 4th, and 5th levels. They are all dispatched on the same frequency, and worded exactly or very similarly to this - "This is MABAS Division X to all locals, the Nice Fire Department has requested MABAS Box 123 to the 3rd level for a structure fire at 123 Rainy St. Staging is at Rainy & Sunny. The following departments are due to the scene. (list them). The following departments are due change of quarters."
The other way it is used, and this is mainly Chicago, but a few other areas as well. Every fire alarm in buildings in Chicago has a box number. Rigs are dispatched to the "Box Alarm" on box 12345. This is correlates to a specific address. Or you often hear, "Main from Engine 89, we're going to be pulling box 12345 at Some School." This for fire drills.
Just how it works here!