What are the units of the band intensities in a western blot image?

The units in your case are arbitrary units.

For instance, you may have taken a photo of the bands with a camera that saved a gray-scale file with an 8-bit depth. That means that the brightness values of each pixel have 256 gradations (or 2^8 values of brightness possible between minimum/blank and maximum/saturated). The program only takes the pixel brightness value anywhere between black (0) and fully white (256) and simply performs the calculations so. There is no real connection between actual band intensity and the output from ImageJ or any software.

Alternatively, you can also have relative units. Say you have a reference: a positive or a negative control. If your reference has a brightness of 10, and your band has a brightness of 7, you could express the band intensity as "0.7 relative units"; here you have to specify your reference, i.e. what you are calibrating your relative units to, hence the term relative.