Hi Eye_3mac
This Thread has been mainly about the Failure of Software Update to do just that.
It seems very likely that most of the ‘failures’ posted were due to the Apple Servers undergoing maintenance.
This created the message that many have included in their posts, IIRC one post carries an image of the message, generated by the Server issue.
Assuming the above is correct it looks like you have something entirely different, although it is unlikely that I can offer any direct help with your problem.
However may I suggest returning to Disk Utility again.
Make sure you have selected your Start Up Disk on the left hand side.
Run Repair Disk.
Run Repair Disk Permissions.
Run Software Update again, and if you still need to read on.
If the Mac crashes again, restart or allow it to restart, then under Applications > Utilities > open Console.app
In the left hand pane look for an entry relating to Software Update Crash this will also have the date and time recorded.
Click on this entry and copy the Log, this will give others an indication of the issues involved.
List everything that may be useful for members who are able & willing to help.
Include your Mac info, Software version and how Software Update is set up, IE. Atomatic or on Demand.
Then start a new Thread with an appropriate Title, IMO ‘Software Update causing crashes’ would seem OK?
Include in your first post all the info and log you have gathered.
Hopefully you will get a meaningful response and be able to resolve your problem.
Good luck.