He was adamant that we don't/didn't hire them.
Is "don't" possible?
If the possibility of hiring them is still in the future, yes. g. someone of a certain nationality, and this is a general statement about what sort of people to hire (at any time).
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If the possibility of hiring them is still in the future, yes.
It is also possible if "them" is a general catgory of person, e.g. someone of a certain nationality, and this is a general statement about what sort of people to hire (at any time).