You can be the product whether you pay or not, and you can not pay and still not be the product (e.g. Wikipedia).

It's a thought-terminating cliché.

Logically, if the converse is false, then the original statement is false, correct?

"If you are the customer, then you are not the product."

This is false (many contradictory examples show that you can be both customer and product), and so the original quote itself is false.

Here's a Venn diagram:

http://imgur.com/532xPbd

Note that "If you're not the customer, you're the product" is true, but "If you are the customer, you're not the product" is false.

Your reasoning would hold if the categories were mutually exclusive, but, as you noted, they aren't.

No, you're thinking of contrapositive. Converses are not equivalent. In this case, contrapositive would be "if you're not the product, then you are paying for it."