This is one of those "Soft people skills" problems that can be hard to navigate. Driving a Honda to work at a Ford factory is a pretty big problem, because it's obvious to everyone that you're doing it: you don't have the ability to just not talk about it.
I would expect your actual problem (who insures your car) should not become public knowledge, though. Insurance decisions are not public information, and even if someone in the company is privy to your lack of insurance with the company, it's not acceptable for them to discuss it in a public setting or reveal that information to another person. This would be a bigger deal if it were HIPAA information (say you worked at Red Plus and Orange Stave, but used Humanoid insurance); that would be likely actionable in court. But even with auto insurance information, there is some expectation of privacy there, as long as you don't reveal it yourself.
However, now it's out, how to deal with it is tricky. You may seem disloyal to the company, after all, and that can be hard to accept for your co-workers. It sounds to me like you handled it well - explaining your reasons for why it doesn't work out for you, but that you like the insurance overall. That's not going to remove all of those feelings, though. You may simply need to accept that some people won't love your decision, and be okay with that. Or change to your company's insurance, of course, but it sounds like you don't want to do that.
I would definitely use it as an opportunity to argue for the improvements that can be made to make it more acceptable to you (unless that's impossible because you're not in the niche your company is trying to sell to). If it's a matter of not having a good pricing structure for people in your income level, for example, use it as a positive: "Right now I can't afford our insurance, even with the employee discount - how am I supposed to sell this to clients?" If it's a matter of a specific issue, such as lack of coverage for rental cars, point out that you're not the only one that isn't getting insurance for that reason - suggest vocally that it be added as an option.
And of course, if you're not in the niche you're selling to - such as you're working for a bargain-basement insurance company and you need more comprehensive coverage for your teenage daughter's escapades - point that out, also.