The US restrictions on travel to Cuba are found in the Treasury Department regulations. The prohibitions generally apply either to "any person within the United States" or "any person subject to the jurisdiction of the United States."
The first term is defined at 31 CFR 515.330:
(a) The term person within the United States, includes:
(1) Any person, wheresoever located, who is a resident of the United States;
Whether this applies to your or your husband therefore depends only on your place of residence, not on your citizenship, but since your husband's legal status is "lawful permanent resident," he might risk losing that status if he tried to assert that his place of residence was outside the US.
The second term is defined at 31 CFR 515.329:
The terms person subject to the jurisdiction of the United States and person subject to U.S. jurisdiction include:
(a) Any individual, wherever located, who is a citizen or resident of the United States;
This would apply both to you and to your husband. Entering Cuba with a passport from a country other than the US does not change the fact that you are a US citizen with respect to US law.
Someone who is subject to the regulations can nonetheless travel legally to Cuba if they have a license from the Office of Foreign Assets Control (OFAC). However, OFAC notes that there are 12 categories of license for which you do not need to apply:
Travel to Cuba
You may not need to apply for a specific license. Before applying for a specific license, please review the information about Cuba travel and the 12 categories of general licenses related to Cuba travel found in the Cuba Assets Control Regulations (31 CFR part 515). To the extent that your proposed travel falls within the scope of an existing general license, you may proceed without applying for a specific license from OFAC. General licenses constitute blanket authorization for those transactions set forth in the relevant regulation and are self-selecting and self-executing. Persons traveling pursuant to a general license do not need to notify OFAC of their travel plans. It is OFAC's policy not to grant applications for a specific license authorizing transactions where a general license exists.
If you determine that a general license does not apply, you may apply for a specific license by using this online application process. OFAC will consider the issuance of specific licenses on a case-by-case basis when a general license provision is not available. Please read all instructions and relevant information fully before submitting an application for a specific license.
The 12 categories, at 31 CFR 515.560, are:
(1) Family visits (see § 515.561);
(2) Official business of the U.S. government, foreign governments, and certain intergovernmental organizations (see § 515.562);
(3) Journalistic activity (see § 515.563);
(4) Professional research and professional meetings (see § 515.564);
(5) Educational activities (see § 515.565);
(6) Religious activities (see § 515.566);
(7) Public performances, clinics, workshops, athletic and other competitions, and exhibitions (see § 515.567);
(8) Support for the Cuban people (see § 515.574);
(9) Humanitarian projects (see § 515.575);
(10) Activities of private foundations or research or educational institutes (see § 515.576);
(11) Exportation, importation, or transmission of information or informational materials (see § 515.545); and
(12) Certain export transactions that may be considered for authorization under existing Department of Commerce regulations and guidelines with respect to Cuba or engaged in by U.S.-owned or -controlled foreign firms (see §§ 515.533 and 515.559).