I have a 19 year old son who is going to college. Can he file taxes, and can we his parents still claim him?

Even a new born must file the tax return, if there is taxable income. For dependent's filing requirement Read: Your U.S. Tax Return: Filing Requirement for a Dependent

He must file his separate tax return.

Yes, you can claim him dependent if you meet the requirements.
A dependent can be your qualifying child or qualifying relative. Read: Your U.S. Tax Return: Requirements for claiming a dependent