Will a universal cell phone charger damage your cell phone battery? ?

I was looking for a cheap cell phone charger for my cell phone. The cheapest one was universal and a friend told me that the universal chargers will damage your cell phone battery. He said that someone from verizon told him that the universal chargers don't shut off when your battery is completely full causing it to damage and decrease durability. Is this true? Because the charger on verizon's website is $30 and the universal is only $10. Anything will help, thank you.