However, he mentioned a coughing colleague who doesn't cover his or her mouth and won't stay home. There are lots of people who do the right thing and stay home. However, there will always be people who can't let go of their jobs or can't afford to take off without pay and will try to hang in there. Maybe more bosses need to send out a memo to employees asking them to please stay home if they have a cold or the flu and offering to pay people without sick leave for their time off if they bring in a doctor's note. In the long run this may save the company money.
Fewer people passing germs around, long term, means fewer people out of work.
You want your boss to see the value of your proposal so you would not want to present it as a demand. Think of ways to make it an outgrowth of something that your boss has discussed with you or others over the previous few months. You want to make sure that your boss is in a good mood when you present your proposal. As they say, timing is everything. Be ready to respond to any questions that may pertain to money.