Will any Digital Converter Box work in place of the Cox Mini Box?

I don't want to rent 3 Mini boxes from COX for the foreseeable future.  That's $9 per month forever.  So $108 for the first year alone.  This is the exact reason that I moved away from Direct TV!!  Will any digital converter box work on my TV's which are currently connected directly to the Coax Cable?  I've asked the COX people several times and the all say the same thing "I am not sure" which to me sounds like "Yes, but I can't tell you that because we'll lose out on your $110/year".  Wal-Mart sells the Ematic AT103B Digital Converter Box for only $32 and it has a built in DVR!!  Can someone please help me to answer my question???