Pass parameter from VbScript to vba function

In your VBScript xlObj is set to be an application Set xlObj = CreateObject("Excel.Application"). That means xlObj.Application should be xlObj only.

In your VBScript Filename is not declared nor set to a value therefore it is empty. You need to define value to it.

Set xlObj = CreateObject("Excel.Application")
Set objWorkbook = xlObj.Workbooks.Open("E:\Headers.xlsm")

xlObj.Visible = False
xlObj.Workbooks.Add

Dim Filename 'declare filename and set a value to it
Filename = "E:\YourPath\Yourfile.xlsx"        

Dim Result
Result = xlObj.Run("Headers.xlsm!Headers", Filename)

xlFile.Close True
xlObj.Quit

In your function you use Exit Function. This will stop the code immediately at this point, which means your workbook myWb will not be closed! It stays open because myWb.Close is never reached. Change Exit Function to Exit For to just exit the loop and continue to close the workbook.

Cells(1, i).Value is neither specified which workbook it is in nor which worksheet. This is not very reliable never call Cells or Range without specifying workbook and worksheet (or Excel will guess which one you mean, and Excel can fail if you are not precise).

Therfore I recommend to use something like myWb.Worksheets(1).Cells(1, i).Value if you always mean the first worsheet in that workbook. Alternatively if it has a defined name using its name would be more reliable: myWb.Worksheets("SheetName").Cells(1, i).Value

If you turn off ScreenUpdating don't forget to turn it on in the end.

Error handling in case filename does not exist would be nice to not break the function.

You can slightly improve speed by assuming Headers = "True" as default and just turn it False in case you find any non matching header. This way the variable is only set once to True instead of multiple times for every correct header.

Public Function Headers(ByVal Filename As String) As String    
    Application.ScreenUpdating = False

    Dim flag As Boolean 'flag is never used! you can remove it

    On Error Resume Next 'error handling here would be nice to not break if filename does not exist.
    Dim myWb As Workbook
    Set myWb = Workbooks.Open(Filename:=Filename) 
    On Error Goro 0 'always reactivate error reporting after Resume Next!!!

    If Not myWb Is Nothing Then            
        Dim Arr() As Variant
        Arr = Array("col1", "col2")

        Headers = "True" 'assume True as default and just change it to False if a non matching header was found (faster because variable is only set true once instead for every column).
        Dim i As Long 'better use Long since there is no benefit in using Integer
        For i = 1 To UBound(arr) + 1 'use `ubound to find the upper index of the array, so if you add col3 you don't need to change the loop boundings
            If Not myWb.Worksheets(1).Cells(1, i).Value = Arr(i - 1) Then 'define workbook and worksheet for cells
                 Headers = "False , Not Found Header " & Arr(i - 1)
                 Exit For '<-- just exit loop but still close the workbook
            End If
        Next i
    Else
        Headers = "File '" & Filename & "' not found!"
    End If

    Application.ScreenUpdating = True
    myWb.Close
End Function