Prevent Formula #REF! When VBA Macro Deletes Columns

I'm using the following code to delete columns:

Dim rng As Range
Dim i As Integer, counter As Integer

Set rng = Range("1:1")

i = 1

For counter = 1 To rng.Columns.Count

'If cell i in the range contains an "x",
'delete the column
If rng.Cells(i) = "x" Then
rng.Cells(i).EntireColumn.Delete
Else
i = i + 1
End If

Next

My problem is that I have cells in other worksheets linked to the worksheet that is running this macro and everytime I run it, I get a handful of "#REF" errors. I think this problem might be solved if I could simply delete the contents of the column rather than deleting the entire column. How can I modify my code to do this?