There are two problems I see with your script.
- The variable "$user" doesn't exist.
- The 2nd line of the script has no double-quotes surrounding the data
If you simply add the necessary quotation marks you'll still have a problem because the string will be placed into the "success stream" (i.e. the "pipe"). You don't want that to happen (at least not in this case).
Also, the use of "$($.EmployeeID)" instead of the simpler "$.EmployeeID" isn't necessary as the expression isn't surrounded by double-quotes. [Ignore the missing 'underbar' following the "$" in this paragraph. It's an artifact of the editor when posting code as text!]
Try this:
Import-CSV C:\EIDUpdateTEST1.csv |
ForEach-Object {
Write-Host "EmployeeID $($_.EmployeeID)"
Get-ADUser -Filter "EmailAddress -eq '$($_.EmailAddress)'" |
Set-ADUser -EmployeeID $_.EmployeeID