You can always use larger size wire, however, you may run into a few problems doing so.
Terminal size
The fist problem you'll have, is that the 6 AWG wire cannot be directly terminated at the new 20 ampere breaker. 20 ampere breakers tend to only accept 14 AWG to 8 AWG wire, though this may vary a bit by brand. To solve this, you're going to have to use some smaller wire as a pigtail, to make the connection to the breaker.
You'll run into the same problem at the receptacle, as most 20 ampere receptacles only accept 14 AWG to 10 AWG wire.
Box fill
Since device boxes aren't designed to handle 6 AWG conductors, you're likely going to have trouble fitting all the wires in a normal single gang device box.
If you decide to install a single receptacle, and cap off one of the ungrounded conductors. You're going to need a box that has a volume of at least 24.5 cubic inches.
Three 6 AWG current carrying conductors @ 5.00 cu.in. per = 15 cu.in.
Grounding conductors based on largest conductor = 5 cu.in.
Device @ 2 times largest conductor connected to it (12 AWG) = 4.5 cu.in.
Total = 24.5 cu.in.
To make matters worse, you'll probably be looking for a weatherproof box, since you're installing it outside.
If you want to make the splice in a junction box, and then run 12 AWG to a device box. The junction box will also have to have a volume of 24.5 cubic inches.
Three 6 AWG current carrying conductors @ 5.00 cu.in. per = 15 cu.in.
Grounding conductors based on largest conductor = 5 cu.in.
Two 12 AWG current carrying conductors @ 2.25 cu.in. = 4.5 cu.in.
Total = 24.5 cu.in.
Which means you could use a 4 11/16" x 1 1/4" box for the splice. Then you'd continue to the device box with 12 AWG conductors, which would fit just fine in any device box.
So, can it be done? Sure. Is it simple and straightforward? Not really.