No, they exsist, several actually. Broken up into two types, those with cable cards, and those without. Ones with cable cards offer more channels, but usually cost more. Your not going to find many/any with HDMI in and have a storage device. Thats because HDMI is protected by HDCP and usually won't allow recording. On that note, this is just general info, as I am not sure what the rules are for recording content. I personally have my DCX3400 and love it.
Some common PVR's with cable card are Tivo and Moxi. They range anywhere from 300-700$ I believe. Tivo has a subscription service while Moxi does not(right?). Moxi was also bought out by Arris but you can still find their units on Amazon.
Some common PVR without cable cards are Hauppauge and Avermedia. They range from 150-200$ and differ greatly in capability and available inputs, so watch out. They require a video source though, so you would still need to have some kind of cable box from Cox, just not a DVR.
As for "Boxx" I think you may be thinking of "Boxee" which started as a Dlink product, but moved on to what your probably thinking of
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www.boxee.tv/#homeI looked around though, and it hasn't gotten very good reviews. Seems the firmware is very buggy. On top of that, you have to pay 10-15$ for the Cloud storage "Unlimited DVR" feature, so that makes it kind of pointless as a DVR alternative.
I think a example of the kind of thing your looking for may be
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www.newegg.com/Product/P ··· 15100097