I think that I finally found a solution to this issue which I discovered more than a year ago. I am interested to hear any feedback from other users that experienced this issue, in as far as I am aware, Quicken support never made any acknowledgment of this issue, nor recommended any ways to work around this.
My first attempt to eliminate this issue was to go into the budget view and "touch" every "Everything Else" category, which was always zero in my case. This eliminated most but not all of the Everything Else categories appearing on budget reports.
The next thing I did, which seems to have eliminated the problem completely for me on budget reports, was to go to "Manage Budget Categories" and un-select all of the parent categories that appear above the sub categories. Since I don't budget amounts for the parent categories, there was no loss of information for me on budget reports.
But, this seems to have cleared up the issue of "Everything Else" showing up on budget reports.