Can I sue the a business for an oil change that it perform on my car which resulted in damamge to it?

You can sue the oil change place for negligence.  Negligence is the failure to exercise due care (that degree of care that a reasonable oil change business would have exercised under the same or similar circumstances to prevent foreseeable harm).

In order to prove negligence, you will need to prove duty (of due care mentioned above), breach of duty (failure to exercise due care), actual cause, proximate cause, and damages.

Actual cause means but for the oil change business, would your engine have been damaged?  If the answer is no, which appears to be the case, actual cause has been established.  Proximate cause means are there any unforeseeable, intervening events which would relieve the oil change business of liability?  If the answer is no, proximate cause has been established.  Damages means the amount of compensation you are seeking in your lawsuit for negligence.  Your damages would be the cost of replacing the engine.  Your damages should also include the cost of a rental car.  You will need to mitigate (minimize) damages by selecting a rental car with a reasonable rate.  If you were to select the most expensive rental car you could find, you would have failed to mitigate damages and your damages would be reduced accordingly.  The same applies to the engine replacement.  You would need to have a comparable replacement engine in order to mitigate damages installed by a mechanic whose charges are comparable to other mechanics in the area performing engine replacement.  If you were to select the most expensive mechanic you could find, you would have failed to mitigate damages and your damages would be reduced accordingly.