The short answer is NO. What employees do after work if there business as long as they don't engage in illegal or immoral activities that would case a bad light on their employer. Additionally, some jobs require employees to sign a contract and agree to be governed by highers standards (i.e. TV news people may have different standards then the person working at McDonalds and professional athletes have to conduct themselves in various ways like wearing suits when traveling etc. which are all bargained for in the contract).
This type of coercion may also be a violation of various labor laws as well. However, as always, you have to decide whether or not you want to fight this battle -- suing because you lost your job is not an enviable position. It may be hard to find counsel. And it may take a long time to get to case resolved. In the meantime you're not earning money. You always need to keep this kind of consideration in mind before deciding to take on an employer. Your situation involves important and complicated matters of law and fact. I urge you to consult with a lawyer on this matter.
Under Illinois law these restrictions may be upheld and YES in some cases the Illinois employer can tell their employee's what to do on their own time. Illinois is a pro-employer state, with most of the laws giving the upper hand to the employer. An employer in an at will situation (no union or other contract) can set up rules or policies that the employees must follow. Illinois courts only look at if the rule or policy is discriminatory by age, sex, race, ethnic origin, religion, nationality, disability. Illinois follows the discrimination bases set up by the eeoc (www.eeoc.gov).
There have been cases where a policy about not smoking outside of the work place (on the employee's own time) was upheld. All the employer would need to do would be to convince the court that allowing the behavior could hurt the employer. If the people who are currently employed want to fight this they certainly would need an attorney. However, in this poor economic times it is doubtful that anyone would want to roll the dice on their job.
As for you, in Illinois one of the common sections in a severance agreement is that the employee who received the severance cannot tell current employees of the same employer about the severance agreement. Illinois courts routinely uphold this because the employer could be harmed if all employees expect to get the same severance. So you need to look at your severance agreement for the same type of wording. What the employer is attempting to avoid is the current employees hearing what a good severance you got then insisting on the same dollar amount. It is common for employers to go after a former employee who "leaked" info about their severance to people who still work for this employer. The employer finds out when the current employee demands the same or larger dollar amount.