Though I've never hunter them, I've talked to many people who have and, like many have said, usually a big gun is recommended. I once ran into a gunsmith at a local range who was test firing .338 Utra Mags he had built for nilgai-hunting customers. And I've heard of guides who won't allow anything less than a 300 mag.

I suspect if shots were taken thru the shoulder instead of behind it, the gun recommendations wouldn't be so large. However, remember that the people making these recommendations have seen more nilgai shot than most people have seen nilgai.
My advice would be to trust those who have BTDT.

Only a fool would sell an accurate .30-06