I think it is totally possible that there's a more "complete theory" in the UV with a positive mass squared term that, under the RG flow, the mass term turns negative in the IR (e.g. at the electroweak scale).
For example, in this paper, the authors show a similar scenario: the Higgs mass squared term at high energy is originally positive. In the RG flow, the mass term receives contributions from various couplings and becomes negative at lower energy scales.
This cannot happen in the standard model, but can happen in SUSY theories.