Wow, I am disgusted to here that an employer would do this. This is not legal of your employer and pretty downright unprofessional.

Do NOT pay the invoice.

He sounds like a scam. What I suggest is that you demand a 1099 from him. The 1099 is a tax form that an employer sends you when you do not pay quarterly or per-pay taxes. A 1099 is the legal way to go when you have not paid taxes during the year. I am not an accountant, but I am answering this because I witnessed a friend going through a similar situation. Also, I, too, had a job where I was paid checks with no taxes withheld. The 1099 is what MY accountant suggested and when I filed it, I paid the taxes estimated from the gross amount on the 1099 form.

What your employer is doing is
1. saving money paying by not paying quarterly employment taxes on your labor.
2. trying to make more money off of you by invoicing you for the taxes that you should be paying the government. I am sure he has no intention of paying, but probably just pocketing.
3. Saving money by not hiring an accountant of his own.
4. Getting YOU into serious trouble with the IRS.

All 4 of these are illegal. Demand the 1099 and if he refuses, threaten legal action. Also, hire yourself an accountant.

I hope this helps,
Bruce