What would happen if told border patrol that my boyfriend was unemployed?

There's no reason to mention self employed, you can just put "Painter" as the job title. What CBP would mainly want to know, if at all, is if your boyfriend intends to work while in the USA or if he intends to overstay his visa. To help strengthen his case, have proof of a return ticket to provide them with. That can help show intent in case your boyfriend gets questioned. If he has any paycheck receipts, or university/training certificates he could bring copies of those, just in case. Most people traveling on tourist visas do not usually bring such things.

Keep in mind, on the Visa Wavier Program he can stay up to 90 days for tourism or business purposes.

Overview

The Visa Waiver Program (VWP) enables nationals of certain countries to travel to the United States for tourism or business (visitor visa purposes) for stays of 90 days or less without obtaining a visa. Not all countries participate in the VWP, and not all travelers from VWP countries are eligible to use the program. VWP travelers are required to apply for authorization though the Electronic System for Travel Authorization (ESTA), are screened at their port of entry into the United States, and are enrolled in the Department of Homeland Security’s US-VISIT program

In the comments you mention your boyfriend is a house painter. FYI, there is a O'NET job code for this 47-2141.00 - Painters, Construction and Maintenance career, identified by the US Department of Labor and Training Administration. So it s definitely a known career field.

47-2141.00 - Paint walls, equipment, buildings, bridges, and other structural surfaces, using brushes, rollers, and spray guns. May remove old paint to prepare surface prior to painting. May mix colors or oils to obtain desired color or consistency.

Sample of reported job titles: Facilities Painter, Foreman, Highway Painter, House Painter, Industrial Painter, Journeyman Painter, Maintenance Painter, Painter, Painter Foreman, Senior Painter