WORKING!!! Thanks so much! The help has been wonderful -- a lifesaver.
Here's the final version:

Sub OpenMultipleUserSelectedFiles()
Dim rng1 As Range
Dim rng2 As Range
Dim tempBk As Workbook
Dim dataBk As Workbook
Dim filearray As Variant

filearray = Application.GetOpenFilename _
("Text Files (*.*),*.*,PRN Files (*.prn),*.prn", , , , True)
If IsArray(filearray) Then
For i = LBound(filearray) To UBound(filearray)
Workbooks.OpenText filearray(i), _
Origin:=xlWindows, StartRow:=1, DataType:=xlDelimited,
TextQualifier _
:=xlDoubleQuote, ConsecutiveDelimiter:=False, Tab:=False, Semicolon:= _
True, Comma:=False, Space:=False, Other:=False, FieldInfo:=Array(1, 1)

Set tempBk = ActiveWorkbook
With tempBk.Worksheets(1)
Set rng1 = .Range(.Cells(1, 1), .Cells(Rows.Count, 1).End(xlUp))
End With

On Error Resume Next
Set dataBk = Workbooks("data.xls")
On Error GoTo 0
If dataBk Is Nothing Then
Set dataBk = Workbooks.Open("C:\MyTextFiles\data.xls")
End If

With dataBk.Worksheets(1)
Set rng2 = .Cells(Rows.Count, 1).End(xlUp).Offset(1, 0)
dataBk.Save
End With

rng1.Copy
rng2.PasteSpecial xlValues, Transpose:=True
tempBk.Close Savechanges:=False
Next i
Else:
MsgBox "You clicked cancel"
End If
End Sub