I have not read all the responses, but I do have a suggestion for the OP. Instead of trying to get your boss to increase the number of sick days, my suggestion would be that you ask him to consider revamping your traditional vacation/sick plan to a PTO (paid time off) plan. PTO can be used for sick OR vacation, giving you more flexibility with your time off. If this isn't an option, perhaps asking if you can carry forward your sick leave from one year to the next, would allow you to build up some time if you had a longer term illness. If that isn't an option either, then find yourself a short term disability policy (which I would recommend regardless of the plan you have at work, but since you work in the insurance industry you probably already know this). Finally, I assume you can use vacation tome when you are sick, if you run out of sick time.
I work in government, for many years we had the traditional sick/vacation plan, although vacation would increase with years of service, our sick leave did not, and that is not an unusual pratice in ANY industry. We happened to receive about 12 days of sick leave per year, no matter how long you had been there. Many people, like me, had oodles of sick leave that we would likely never use. When we transitioned to PTO, all new employees had to have that plan, but exisitng employees had the option of keeping the traditional plan, or going to PTO. I went to PTO.
The conversion to the PTO plan meant that I lost a bit more per month, in terms of total hours, than if I had kept the traditional plan, but I gained oodles of flexibility in how I use my time. In February I will have been with my employee for 20 years, and my PTO goes to 11.23 hours accrued per payperiod, so I will be earning more than an employee with 20 years of service who has the traditional sick/vacation plan. I will have about 7 weeks of time available to me each year. With our PTO plan, we can carry forward 500 hours from year to year. I currently have about 400 hours on the books (I also have about 400 hours of sick leave that I carried forward into my PTO plan).
I am a supervisor, and it has been amazing to watch how people manage their time when they have PTO. They naturally save their time for vacation, and minimize the time they call in sick. Employees who continue to have the traditional sick/vacaation plan, take more sick time. I guess this leads to another discussion about people calling in sick when they are not, but I promise I won't go there and clutter up this thread with a big debate!