Is it illegal to fire somebody with a doctor’s note and on their day off

Did you have and use paid time off (e.g. sick days) you had earned to cover any absence? If you did, you may not be terminated for using a benefit (sick days or PTO) you had worked for and earned, and if you were terminated for using PTO, you may have a wrongful termination claim: consult with an employment law attorney about possibly suing and what you might be entitled to.
If you did not use PTO for any absence, you may be terminated: the law does not require employees to retain employees who miss work other than with the use of PTO (or FMLA leave, if you and your employer were both covered and this absence was eligible for Family and Medical Leave Act leave--you can find the rules for FMLA leave on the U.S. Dept. of Labor website). 
You may also be terminated for reasons unrelated to the infection, like poor performance, insubordination, other absences or lateness, etc.
A doctor's note does not protect you--it is widely believed that one does and that you can't be terminated if you have a doctor's note, but that is not true. All a doctor's note does is provide proof that you were sick, but it does not itself give you the right to miss work.
You may also be terminated on your day off: there is no legal requirement that it be done when you are at work.