I have a Directv tivo unit (nonHD) about two years old. If I were to get a directv HD unit (they are offering them for free) non dvr could I hook the two together?

Really it depends if the dvr is the new r15 model from direct tv you definitely cannot seeing as the r15 does not have any type of input. Some of the stand alone tivo units do have inputs. Your best bet would be to get a dvd recorder, or direct tv is offering a deal right now to get the Ka/ku 5 lnb dish and a Hr-20 the hd dvr that does it all for 199 after a mail in rebate