Short answer: No, Jews cannot eat non-Kosher ("unclean") animals. Even kosher animals must be properly slaughtered and have all blood removed.
Long answer: As mentioned above, context is very important. Using an easy-to-understand translation and emphasizing all the key words is also important. "...the unclean and the clean may eat thereof..." "May eat thereof" are the important words.
Here's that verse, in context in the New JPS translation (which should be easier to understand):
10 When you cross the Jordan and settle in the land that the Lord your God is allotting to you, and He grants you safety from all your enemies around you and you live in security, 11 then you must bring everything that I command you to the site where the Lord your God will choose to establish His name: your burnt offerings and other sacrifices, your tithes and contributions, and all the choice votive offerings that you vow to the Lord. . . . 13 Take care not to sacrifice your burnt offerings in any place you like, 14 but only in the place that the Lord will choose in one of your tribal territories. There you shall sacrifice your burnt offerings and there you shall observe all that I enjoin upon you. 15 But whenever you desire, you may slaughter and eat meat in any of your settlements, according to the blessing that the Lord your God has granted you. The unclean and the clean alike may partake of it, as of the gazelle and the deer. 16 But you must not partake of the blood; you shall pour it out on the ground like water.
As you can see, these verses are describing the offering of sacrifices and the consumption of meat. Prior to the building of the Temple in Jerusalem, when an Israelite wanted to eat meat (which usually was only on special occasions), he would offer an animal as a sacrifice to God on his "backyard alter," (במה\במות) and partake of the remainder, thus including God in his meal. (Source.) These verses prohibit the use of "backyard alters" once the Temple has been built. Since sacrifices are restricted to the Temple, if an Israelite wished to eat meat, he no longer has to offer it as a sacrifice. "But whenever you desire, you may slaughter and eat meat..."
Only ritually "clean" or "purified" Jews who had immersed themselves in the mikvah were allowed in the Temple, whereas ritually impure individuals (such as one who came in contact with a dead body) were prohibited. One might think that only ritually pure individuals would be allowed to eat meat. Verse 15 teaches us that this is not so, "the unclean and clean alike" can eat from the meat.
Why does the verse mention the gazelle and the deer? Rashi explains that they're mentioned to illustrate that the logical extension of the above holds true. Gazelles and deer, while kosher, are not permitted as sacrifices in the Temple. But since the Israelite no longer has to offer the animal as a sacrifice before eating them, he is now permitted to slaughter and eat the gazelle and deer.
The above was the "simple" reading of the verses. Halakha, or Jewish Law, reinterprets them slightly, due to the odd mentioning of the gazelle and deer, and the fact that this basic idea is repeated in verse 20. The halakah explains that these verse refer to a specific case of an animal that was designated for a sacrifice and then became blemished. For the details of this approach, please see Rashi's commentary on the verse, available in English here.