As a songwriter who writes Country songs and records the demos as a guitar/vocal, this question is right up my alley.
Country music is characterized by both the subject matter of the lyrics as well as the instrumentation, style of play, and vocal style. Although modern country songs topping the country billboard charts today have incorporated many elements of pop and rock with both instrumentation and arrangements.
To add a little extra beyond the basic question - for the edification of other readers -
the most common instrumentation associated with more traditional country music is acoustic guitar, Fender Telecaster electric, pedal steel, drums and often a banjo or fiddle for the more blue grassy end of the country spectrum.
Chord progressions in traditional country are usually pretty simple with few if any augmented or diminished chords. In fact there is a running joke among musicians in my area that all you need to play any country song is a G a C and a D chord (although sometimes I get fancy and through in an Em for good measure).
From a subject matter point of view, Country seeks to speak to the hopes, dreams, triumphs and tribulations of the common man in plain straightforward conversation like language that is easy to understand. Country lyrics don’t often contain deep metaphors with hidden meanings or clever plays on words.
But enough about country music in general. Let’s talk about your specific question. So you have written some lyrics that fit within the permissible confines of what defines “Country music” and you have a chord progression and melody in mind. Now you want to record a simple Guitar/Vocal demo and have it "sound country".
First I would recommend an acoustic guitar if there is only one guitar in your recording. If using electric - use as clean a signal as you can get.
To give your guitar playing more of a country feel, you can employ any of several techniques. I find that the easiest to use is a bass strum bass strum pattern with a simple bass walk or hammer-on between chords. I call it an alternating bass pattern and play it with a pick. In it's basic form the strumming pattern is Bass down/up Bass down/up in 4/4 time.
If you want to get a little more complex and have the skills to pull it off, Travis Picking – a fingerpicking technique made popular by Merle Travis, will lend a country flavor to any guitar arrangement. There is a great lesson here Travis Picking Lesson that has examples with tab, standard notation and sound files that you can adjust the playback speed making the examples easy to learn regardless of your preferred learning style. This is different than "Chicken Pickin" another country style that features many muted plucked notes (often using both pick and fingers) but Chicken Pickin won't work as well in a solo guitar arrangement with no other instrumentation.
As far as vocal style for country, to be completely convincing, you almost need a country accent. If you don’t have one, please don’t try to fake one. Real country folk might think you are making fun of them ;).
But you can countrify the vocals by adding a little twang here and drawl there. Draw out some of the phrases and try scooping out some phrases by dipping the pitch of the sung note/s and raising it back up. Don’t try to sound too proper in your enunciation or diction.
You really should listen to some country music from singers like Garth Brooks, Allen Jackson, Merle Haggard, Keith Whitley, Dwight Yoakum, Tim McGraw, Hank Williams and Brad Paisley (just to name a few) - to get a feel for the singing style. You can find good examples on YouTube.
As you listen to some country – especially the older traditional country, you will sometimes hear what I call a country hiccup that sounds like you were singing a note and accidentally hit a turkey call and made a little yelp. That’s one technique you can try to throw in occasionally if you can get it down with practice.
Try to use some nasal accents every now and then to add more country flavor. To get in touch with your nasal voice, try holding your nose while singing and try to get the sound to come more through your nose than mouth. If you connect well with the nasal voice you will feel your nose vibrating when you sing while pinching your nostrils shut. In more technical terms this is more like singing through your facial mask.
This video of The Possum (George Jones)George Jones Bartender's Blues YouTube is a good example of the country vocal sound with some nasal twang and vocal scoops.
Here is another video showing a guitar vocal with a distinct country flavor. Garth Brooks Cover YouTube
If trying to sound country just doesn't come natural to you and it sounds too contrived, and you insist on capturing your idea with true country authenticity, consider hiring a demo singer who sings country. You can listen to them on line and many will actually collaborate via the internet where you can e-mail them the music and they add the vocals and send it back. Here is an article about how to hire a demo singer for your country song. Hire a Country Demo Singer The cost for this is fairly reasonable.
Listen to some examples of country singers and just do your best to emulate them. If you want to record the guitar and vocal yourself just for fun, don't worry about who you might offend or if you sound authentic - and just have fun with it. You can always say it's just a spoof. Good luck.