USC sec. 7701(a)(16) Withholding agent.
"...means any person required to deduct and withhold any tax under the provisions of sections 1441, 1442,1443, or 1461."
1441 - Withholding of tax on nonresident aliens.
1442 - Withholding of tax on foreign corporations.
1443 - Foreign tax-exempt organizations.
1461 - Liability for withheld tax.
USC Sec. 3402(a)(2) Income tax collected at source.
(1) "In general, except as otherwise provided in this section, every employer making payment of wages shall deduct and withhold...."
USC Sec. 3402(m) Employees incurring no income tax liability.
"Notwithstanding any other provicions of this section, an employer shall not be required to deduct and withhold any tax under this chapter upon a payment of wages to an employee if there is in effect with res[ect to such payment a withholding exemption certificate..."
The W-4 is a "Withholding Allowance Certificate" It is different than a "withholding exemption certificate"
Seems that by law, an employer is not required, if certain criteria are met.
They are wrong in telling you that you don't have to file. That determination is to be made by you.