I don't see how this thread can move forward without becoming political - social housing in the US, commonly described as "Section 8" housing, "public housing," or "rent controlled housing," will have different rules based on the ownership structure and their ability to enforce those rules vs. their local/state jurisdictional laws. In general, Americans in this kind of housing, unless otherwise prohibited, are protected in MOST states from corporate/landlord policy which would restrict their rights. Some landlords impose rules which slide under the legal radar, but would not survive contact with a court of law if a tenant fought against imposition of the rule.
Pretty plainly, gun rights in the US are not the same as gun laws in the the EU.
Personally, I think it's absolutely asinine to suggest that low income Americans should have their inalienable rights, as guaranteed by their Constitution, infringed simply because of their low income status. So no, having a low income and living in social housing should not disqualify or restrict an American from their right to keep and bear arms, and no, I would not support any rule or law which would infringe upon their rights by assigning this housing as a "gun free zone." Such restriction is fascist elitism, classism, and unconstitutional for any American which has not otherwise surrendered their right to bear arms as a prohibited person (such as convicted felons or those who have renounced citizenship).