A trace could be a big help here. I looked a bit closer and it seems like the logic you use to check the check boxes is not he same as when you manually check the box then update the record. If you run the trace, you might see that when you click a checkbox and then update in the BAQ, the trace might show a few methods being triggered beyond just the Update method. If this is the case, then in your button logic, you may have to check firm and change job num for each line. But here we are getting into duplicating the efforts of well crafted BO methods.
This seems like it should be doable through a simple update BAQ, but the reality may be a bit more complex under the hood. Look into tracing. And if you haven’t use Jose’s tracing tool to help decipher the huge trace files: