Your safest way to form a petition is to, not do it yourself.
Is there a Union at your hospital? Any union?
If not, find one and confidentially speak to them. They can offer advice and give assistance.
A petition, started by you, sets employees up as potential targets....but, if a union does the petition, employees have some rights and protections under the law.
Face the facts though. Hospitals are businesses and if the similarity in names suggest John Stroger owns the hospital. How is that a problem, who he hires? If he hires a dumb friend, his loss.
Hospitals have to balance profit with patient care, a hard job. Certainly an argument for Socialized Medicine...but are there reasons for cutbacks.
Are you doing your job well, are you efficient? What about your fellow employees and Managers? Were the clinics making money?
Be careful...the Union can help with the petition and protect you.