You might want to check out OklahomaWolf's stereo teardown articles. He's done a lot of the modern stereos, and if it comes to having to having to take the stereo out and taking it apart (which I have a creeping suspicion it will, I have never seen a car stereo with a manual eject button. Doesn't mean they don't exist, just means I haven't ever seen one).
Some have a faceplate button-press sequence to 'force' an eject, if the disc carriage is able to mechanically do so and the control electronics are intact.
Some players 'forgot' that a disc was loaded (the display didn't show the disc icon, but a disc was in the machine) and I had to manually turn the loading motor shaft to unload the disc. The CD player worked fine after that.
Sub-standard quality, warped or paper-labels on discs are a bad idea.