No, it isn't. The illegals do not pay a "22% tax" as you incorrectly claim. They pay prices that are higher than they might otherwise be (whether 15, 20, 25% or other) but the only taxes paid are on the profits from the seller times his marginal tax rate. If that price is higher by 22%, the only amount paid as tax by the illegal in that transaction is not 22% but the merchant's profit times his marginal tax rate.
In several past posts that has been shown to be far less than the 23% FairTax rate that would be paid with the FairTax. These illegals will benefit just as will everyone else by reduced prices when income taxes are done away with, but they will still pay more in tax than they do at present when they pay almost nothing and easily evade the income tax.
Seven friggin years and you are still clueless as to what embedded taxes are. Embedded taxes are much more than the tax on the retailers profits. They are all the federal taxes of workers and businesses that played a part in production and distribution of the product. That is how Jorgenson came up with 22%. You keep playing hokey pokey with what embedded taxes are to fit whatever phoney analysis you are working with at the time.