Hey DeeDee1961,
Welcome to Apple Support Communities.
We see you're experiencing an issue with CarPlay in your vehicle, and you've spoken with the car dealership about it. Were you able to confirm the firmware/software for the stereo system is up-to-date? If not, we recommend contacting the car manufacturer(stereo manufacturer) to confirm.
When a device such as your iPhone receives software updates, app developers and manufacturers of their products also have to ensure their product remains compatible and may release updates as well.
In regards to the message "this accessory is not supported", "this alert might appear when your accessory is damaged, isn't Apple-certified or supported by your device, or your device has a damaged or dirty port. Learn what to do next." In this article, there a troubleshooting steps that may help: If an 'Accessory may not be supported alert' appears on your Apple device. We've also provided the steps below:
1. Update to the latest version of iOS
Some accessories might require the latest version of iOS. Make sure that you update to the latest version of iOS.
2. Make sure the accessory is compatible
If this is the first time that you tried to use the accessory, make sure that it works with your device. Some accessories only work with certain models of iPhone, iPad, or iPod touch. If you aren’t sure, contact the manufacturer.
3. Check the connections
Disconnect and reconnect the accessory to clear the alert. If the alert continues to appear, check the connectors on the accessory and your device for damage or debris. If necessary, gently remove the debris and clean the connectors on your device.
4. Learn more
If this alert appears and you don’t have any accessories connected, contact Apple Support.
Learn what to do if your device won’t charge.
If you used the iPhone 11 in a different car, and you didn't have an issue, we suggest working with the manufacturer of the Chevy Equinox in regards to the stereo.
If your iPhone wasn't used in a different vehicle, but you have another one to test(friend or family member), try that. If you do not have another vehicle, but you do have another iPhone(friend or family member), test it in your 2023 Chevy Equinox. These steps will help isolate the issue further by letting you know if the issue is happening with the vehicle itself or the iPhone.
Hope this helps. Take care.