Euler characteristic can be defined in different ways. This definition works for polyhedra. Personally I found it easier to understand $χ$ through the Betti number definition $χ = b_0 - b_1 + b_2$, where $b_0$ is the number of objects, $b_1$ the number of "handles", and $b_2$ the number of cavities, i.e. "enclosed holes".
Toriwaki & Yonekura found out you can calculate the $χ$ of the image by summing up the $Δχ$ of each 2 * 2 * 2 voxel neighbourhood in the image. $Δχ$ describes how much a part contributes to or alters the $χ$ of the whole.
$Δχ$ can also be used to estimate how much the image stack contributes to the $χ$ of the object its cut from, i.e. you assume that your sample is only a piece of larger whole. Each voxel lining the edges of the image stack affects connectivity. Thus BoneJ first reports $χ$ as if your object were independent and floating free in space. In effect the stack is treated as if it was two voxels larger in each dimension. Then it displays the $Δχ$ value that you can use to estimate how well connected this sub-sample is to the whole. See Odgaard & Gundersen for more details. Unfortunately, to add to the ambiguity, what other implementations of the algorithm often report as $χ$ is actually $χ + Δχ$.