Trump has raised the Birther issue with respect to Ted Cruz, his political rival. I doubt this will have any traction. I don't think Birthers objected to Obama in the first instance because they questioned his eligibility; rather, I think they questioned his eligibility because they opposed him on other grounds.
So even if the legal technicalities are similar, I doubt Birthers will raise the same issue with respect to Cruz unless they have ideological objections to Cruz. Birthism is the consequence of opposing certain candidates, and not the cause. It's a perceived way of derailing an unacceptable candidate.