I would check your state law, but I believe insurance companies can no longer tell you where you can get your car fixed, they can say how much they are willing to pay to get it fixed, since they use a standard scale and some insurance companies will allow non-OEM part or junk yard parts. But you can always argue against that especially if your car is in very good shape
Some of this does depend on your insurance company, If you have go insurance they may stand behind you and let you get it fix any where you like, and not dictate what kind of parts can be used. This is what I have so if someone hits me and their insurance is crap I would work through my company pay the deductible and let them deal with the other company. Now it the other company has the same type of coverage as mine then I work through them get it fix and not worry about paying the deductible.
Just recently, someone hit my car in parking lot the woman admitted fault. Her insurance call me, talk it over, they said I can go to any place I like to have the work done, just send them the estimate. Did so and they cut me a check and it was done with in a week. When back haggled a bit with the place and got the work done for less and had couple other items fixed at the same time. Now if the woman insurance gave me any crap I already arrange just to go through my insurance and let them deal with the hassle.
This is way it pays to go through a good insurance company, because of the nightmares you hear about are from these cheapo insurance companies who could care less about making sure you are happy. Also, if you have you own agent verse someone online this helps alot with dealing with the crap.