Anyone know the policy for the military taking non-citizens wanting to enlist?

That restriction for joining the military is not due to policy and is dictated by law. Current law allows only those individuals who are U.S. citizens or those non-citizens who have a U.S. Permanent Resident Card (i.e., Green Card).

The individual you described is likely a covered DACA recipient, but DACA recipients are not permanent residents. For all intents, they are in limbo - They are not here lawfully, but they are not subject to removal because their status is covered by an (now vacated) Executive Order, not law (although the EO was vacated, there have been a number of judicial decisions regarding what can and can't be changed*).

There are additional ways forward for them in getting a Lawful Permanent Resident (LPR) status other than getting married. However, eligibility for the programs will vary as well as the likelihood of success.

I would highly recommend they talk to an immigration attorney to determine their possible way ahead in obtaining LPR status, but one of the first things they should do is apply for Advance Parole from DHS so they can leave and re-enter the U.S. lawfully. Without having a legal entry into the country, in most cases they will be prohibited from getting LPR.
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* Legal status of DACA - https://www.uscis.gov/sites/default/files/document/legal-docs/TexasII-Dkt-729-SupplementalOrderofInjunction-9-13-23.pdf
* Advance Parole - https://www.uscis.gov/i-131
* H1B Visa - https://www.uscis.gov/working-in-the-united-states/h-1b-specialty-occupations

I remember back in the 60's that one of the airmen at our transmitter site was a Norwegian citizen. I believe he was working toward U.S. citizenship.

They have to be a legal resident to enlist. If they're not, then they can't (well not legally as there has been a couple stories I've seen that people fraudulently enlisted). If he needs citizenship, then he can do it on his own. He doesn't need to get married. I doubt that's any faster and they'll be zeroed in on him to make sure it's real.