I agree that the title is not very clear. However, giving the author the benefit of the doubt, I guess what they're trying to say is "protect my H-1B and [my eventual eligibility for a] green card [that stems from my having an H-1B visa]".

In other words, lose the H-1B, lose the chance to get a green card.

I think the phrasing is just to get the key phrase "green card" in there.

The actual question doesn't mention it, but in the answer she gets into the green card process.