How well does a negative non-contrast MRI of the brain exclude metastasis in a patient with squamous cell carcinoma of the lung?

I don't think the question has enough information to give a good answer. For example, if it was a T3, N2 NSCLC, or a small cell, then "yes" I'd repeat the MRI with contrast. On the other hand, if it was a T1, N0 NSCLC, then "no", I wouldn't.

In other words, if I thought there was a real risk of having brain mets, then I'd order a repeat exam, but if I didn't, I wouldn't waste the patient's money ordering a test I didn't think would likely yield results just for the sake of completeness.

While I don't believe there are any studies for specifically non-con MRI sensitivity for mets of this primary, an MRI should ideally be always performed with and without contrast in the context of cancer. Smaller lesions may not cause significant mass effect or FLAIR change, and not all metastases diffusion restrict or bleed.