Can a private company require employees to provide their personal cell phone number for emergencies or work-related contacts?

For companies operating globally, this is a much larger conversation. There are issues of labor law, privacy and collective bargaining agreements which often limit the employers ability to access employee's personal devices.

Outside of the US, an employer is more often than not required to get an employee's consent for any monitoring and contact outside of work hours is not viewed favorably. In France and Germany, in particular. 

Check the local law in the jurisdiction and have clear policies in place defining what is considered an emergency (with training) if you are planning to adopt such policies outside of the US. 

Generally, yes.  That being said, check your state's laws to ensure you don't trample on any state specific regulations governing the collection/use of employee personal information, including any privacy laws.  Also, depending on the particular state(s), you may have to reimburse employees for reasonable costs associated with employment related use of their personal phones (for example, in California, see Cal Labor Code 2802, https://leginfo.legislature.ca.gov/faces/codes_displaySection.xhtml?sectionNum=2802.&lawCode=LAB)