Can I go to jail for not paying for unauthorized auto repairs?

Typically no--a failure to pay for repairs, whether authorized or not, will not result in criminal liability. The repair shop may be able to sue your son and/or you for the money they claim you owe them, but there is no criminal penalty for nonpayment unless it can be shown that you acted with criminal intent. For example, if they can show that you never intended to pay for repairs but were engaged in an attempt, from the outset, to defraud them (rather than getting into a dispute after the fact about the cost, authorization, etc.), that could possibly lead to criminal liability.

Note that even if the sue you, they don't win automatically--they have to prove the existence of some agreement to pay these costs and that they are legitimate charges, and you can present your evidence and testimony disputing that.