The ASP.NET ListBox web control's AddRange() method expects an array of ListItem which is why it can't so easily be converted from Windows forms.
If you are using the data from the txtBoxNameExternal to completely replace whatever items may already be in the ListBox (as opposed to adding to the items) you can use DataBind() which is probably easiest method:
lstBoxExternal.DataSource = txtBoxNameExternal.Text.Split(vbNewLine)
lstBoxExternal.DataBind()
If you need to keep adding items on each button click there are a few ways you could do this but I usually do it by adding items in a loop:
Dim items As [String]() = txtBoxNameExternal.Text.Split(vbNewLine)
For Each item As [String] In items
lstBoxExternal.Items.Add(New ListItem(item))
Next
Also see this related answer for some tips when binding string arrays to DropDownList/ListBox controls: Binding array of string to DropDownList?