As a guess, I would say that both you and your brother have a program installed that is running when you first boot into Windows ... and this program is causing a memory fault. This might be listed in your Startup folder, or as a Registry call behind-the-scenes. Whatever it is, it is interfering with systray.exe AND the Kernel when you boot, and Windows doesn't like it one bit. A memory fault associated with either of these .dll's will cause programs that normally run in the System Tray to fail to load.
Go to: Start\Programs\Accessories\System Tools\System Information. Go to the toolbar in the new window, click Tools, and look for System Configuration Utility on the drop down menu. This Utility will display a list of all programs that load from the Registry during the boot process. Go through the list, examine all the file paths, and uncheck the boxes for any program that doesn't need to run at startup. You might be surprised at how much stuff is running at the boot ... and 75% of it is unnecessary.
Afterwards, if any problems remain, reinstall Windows to repair any corrupted files.
I suspect this will work, because the identical problem is happening on two different machines. You and your brother share software, don't you?
It would just be too coincidental for both machines to have the same problem, without something in common ... besides a corrupted Win9x installation.
The only other thing I can think of is a virus, and one of you gave it to the other. That might require a re-format, for both you and your brother, if your anti-virus program didn't detect the problem.
That's my two cents.
Good luck ...
Toejam31