Healthy Climate and Family Security Act of 2014 - Amends the Internal Revenue Code to require the Department of the Treasury to establish for 2015 and thereafter a carbon trading program that caps the emissions of carbon dioxide (CO2) from crude oil, coal, and natural gas by requiring crude oil refineries, petroleum importers, coal mines, and natural gas suppliers or processors to purchase carbon permits.

Establishes a declining cap on the quantity of permits issued to reduce CO2 emissions until they are reduced to 80% below 2005 levels by 2050.

Sets forth provisions concerning compliance with permit obligations, issuance of permits for carbon capture and sequestration, and permit auctioning, trading, banking, and borrowing.

Establishes the Healthy Climate Trust Fund and appropriates auction proceeds and penalty payments to it. Returns Trust Fund amounts not used for administrative expenses (as a fraction of auction proceeds) to individuals with a valid Social Security number in the form of Healthy Climate Dividend Payments.

Requires the Department of the Treasury to: (1) impose carbon equivalency fees to be collected by U.S. Customs and Border Protection on imports of carbon-intensive goods, and (2) give payments to exporters of comparable domestically produced carbon-intensive goods.

Requires the Environmental Protection Agency (EPA) to publish a rule listing greenhouse gases (GHGs) that includes anthropogenically emitted (caused by human activity) gases that contribute to global warming but excludes CO2 emitted by the combustion of crude oil, coal, and natural gas and gases attributable to the production of animals for food. Directs the EPA to promulgate regulations to reduce such non-auction GHGs and ensure the regulation of all sources of them within 10 years.