I doubt it would ever become routinely mandatory for everyone, on grounds of simple practicality. Perhaps it might be attempted during an emergency epidemic, but the US now has 300 million people, more flu shots than have ever been produced in a single year, and how is this going to be monitored & enforced?
There are legal precedents for having mandatory vaccinations, such as for polio and the childhood diseases. But those are done continuously on a year-round basis, as children reach certain ages, and then it doesn't have to be repeated indefinitely, so that it's a small number at any given moment. It's not the entire population at once, at the beginning of the flu season each and every year.