What law applies to a worker injured or killed on a land-based drilling rig?

The law of the state where the accident happened. For example, if the accident happened in Louisiana, Louisiana Law would apply.

Can an onshore oilfield worker sue his employer for negligence and recover damages for items like pain and suffering, mental anguish, or loss of enjoyment of life?

As a matter of fact, under Louisiana Law, if your accident happened solely as a result of your employer’s fault your remedy is in worker’s compensation, but if a third party was at fault, you can sue and recover damages from that third party.