The answer is "No, but it is probably commoner to use "which" in non-restrictive (descriptive) clauses and "that" in restrictive (defining) clauses." (I notice that American English seems to do this more often than British English.)
Non-restrictive (descriptive) clauses should be offset by commas; restrictive (defining) clauses are not. However a job is in apposition to “a mechanic” and is merely descriptive. (This is creates a strange, but grammatical, sentence as everyone knows that being a mechanic is a job.)
Your sentence has several forms:
Henry was a mechanic.
Henry was a mechanic, a job. (A job is in non-restrictive apposition to “mechanic” and the comma is necessary.)
Henry was a mechanic, a job,[note the comma] which requires a keen eye. (A job is still in non-restrictive apposition to “mechanic”* but “which requires a keen eye* now qualifies “mechanic” although it is unclear because of a lack of context as to whether the clause is restrictive or non-restrictive.)
Henry was a mechanic, a job that requires a keen eye. that requires a keen eye is a restrictive clause and “a job that requires a keen eye” is a noun phrase in apposition to "mechanic".