Can a visa get you into the U. S army?

No you need to be a Permanent Resident to join.

Only green cards or citizens may enlist.

Only if you qualify for the MAVNI program and the program is open when you attempt to enlist.

Google the program for more details.

Generally, you must be a Green Card Holder or a U.S. Citizen to enlist in the U.S. Army. However, if you are a citizen of Federated States of Micronesia, the Republic of the Marshall Islands and Palau, then you are allowed to enlist, because of international treaties. Or if you are a Canadian with American Indian heritage, you are also allowed to enlist.

By the way, the MAVNI Program (Military Accessions Vital to the National Interest) was suspended back in 2014; you may have to check with the recruiter for the current policy.

I heard they give you papers in exchange for joining the military so...