Took car to oil change shop & had battery replaced. Now car won't start. They said they don't know why & can't fix. What do I do

You may have a potential lawsuit for negligence on the Res Ipsa Loquitur legal theory (cars don't stop working by themselves, ergo someone had to have messed something up, therefore the people who last worked on it must be responsible). Whether or not the oil change shop really did mess something up would be a factual question. I would recommend getting a second opinion on the car from a mechanic before forming any conclusions as to what the problem/damage is and how it originated. Then, I would recommend getting a consultation from a litigation attorney in your area to determine whether (and to what degree) legal action would be a good idea in your case.

The first thing you need to know is why the vehicle isn't starting. Generally speaking, it makes sense that if the car was operating normally before it went into the shop and now it is not starting, the shop might have done something that makes it inoperable. Logic would indicate this to be the case. Unfortunately, that may not be the case. If the vehicle is not starting due to some other factor (i.e.- not due to the battery or the fuel pump), then the mechanic shop would not necessarily be responsible. Somehow, it is important to figure out why the car is not starting now. Then you can determine your rights and remedies.