No it's not illegal to work while you're sick.
There's no law that says you have to stay home from work when you're sick and you can go to work if you're sick with the flu, cold etc as long as you are feeling up to working.
However when I have the cold or flu or other illness I just cannot get myself to go to work because I usually feel so week and tired and lose my concentration.
So it's best to stay home when you have the cold or flu or any other illness especially if you're feeling fatigued because you won't be very productive anyway at work.
Stay home until you feel better then return to work once you get over the illness.