Can they? Yes
If you are at-will employed, unless the out-of-work activity you are doing is specifically protected in your state - your employer is legally able to fire you.
For example in most states with at-will employment, they could fire you for smoking cigarettes (outside of work).
However, it would be illegal to fire you for attending church - as religious freedom is specifically protected.
Will they? No
Firstly, how will they know? Unless you're raving about the "great weekend out clubbing" when you get to the office - nobody is going to be tracking how you're spending your free time.
Secondly, it's very likely to be an idle threat. It's very easy to throw the idea of "don't do this or else" at somebody - but realistically, not many people are willing to go through (expensive) recruiting costs just to make a point. If the boss truly was willing to fire all employees who don't isolate - they would be undermining their own plan of "avoid the office shutting down".
What should you do?
While in the office, just don't mention that you're going out - stay quiet, pretend you just spend the evenings at home, eating food from your stockpile.
But otherwise, although it's easier advice to give than to act on - an employer who feels it's their right to restrict your life to that degree is not somebody that values you at all. I would sincerely consider polishing up your CV and moving at the first chance you have.
Can I be fired for being sick?
One worry that's come up in the comments, is that if you contracted the virus - could they fire you for that?
The answer there is no, at least probably not.
In brief, the FMLA (federal act) gives employees (who work for businesses with at least 50 employees), the right to take off "12 weeks in a 12 month period - because the employee is incapacitated by a serious health condition" (amongst other health reasons).
A serious health condition is one that requires:
inpatient care at a hospital, hospice, or residential medical care facility
incapacity for more than three full calendar days with continuing treatment by a health care provider
I would imagine it is likely that the quarantine imposed on you by health professionals - would fulfill that second criteria, so you would be covered. However, there doesn't appear to be any blanket-rule for COVID-19 as yet - so it may depend on the severity of your infection.