CodedUi method which checks all cells in certain column
Disclaimer: Never done this UITestControl stuff

Edge case

If the searched columnName isn't contained in the header of the table I would expect the returned columnIndex to be -1 and therefor the check for IsNullOrWhiteSpace() will throw an exception.  



Instead of using row.GetChildren() and then access the needed cell by index, you should better use the right tool for the job and call the GetCell(int) method.  



Because an extension method can be also called directly like AllCellsInColumnHaveValue(aHtmlTable, aColumnName); you should ensure that the passed in HtmlTable tableis not null.