I personally think that rhythm is the way to go since a strong harmonic foundation is needed to successfully improvise.
BUT this doesn't mean you can't practice them together! Learn some rhythm for a backing track then attempt to play some lead over it! Just focus on rhythm so you learn to play with the changes instead of against(which isn't terrible but requires much more talent to make it work).
The reason rhythm is better is because it provides the foundation for knowing the "right" notes to solo with. When you are soloing you are not just randomly picking notes from a scale or the neck. You are playing a melody that must fit over the changes(it doesn't always(usually depends on style) have to but it generally sounds better when it does).
Jazz and Country are almost exclusively based on the use of "chord-scale" where the chord is the root for choosing the melody. If you start with learning your scales first, like I did, you might be able to play a billion notes and get some good licks in but it will be harder to make those licks fit over the proper chord. You'll be emphasizing non-chord tones notes too much and while sometimes it can sound cool it generally more than not sounds too random and/or non-stylistic.
When you learn rhythm you'll develop that ability faster and center your thinking around it. When it comes time to solo, it's not much more difficult to do so(at least for some people). The idea is that a "lead" in country and jazz is more about playing notes of the chords rather than of the scales. So when you already know the chords well you simply play them and learn little tricks to use the more "melodic" like(Hendrix did this a lot and you can learn a lot from him).
Also you generally wanna see scales as belonging to chords and not the other way around. This helps knowing which scales go with which chords. When you learn it the other way around you just see a scale and a key without connection to the momentary chord and it can be hard to get used to thinking about it otherwise.
That is, it's generally not preferable(Although it is easier) to think in terms of scales instead of chords. You'll sound better and more like you know what your doing if you think in terms of chords because you'll always fit with the rhythm. Why? Because essentially you are playing the rhythm. If you are simply playing the rhythm then you can't be wrong. When you think in terms of scales though, You have many more notes to chose from and are more likely to get it wrong.
In Jazz and Country there are 4 notes over each chord that are most important. In fact 2 if these are generally regarded as the "key" notes. In jazz it's the 3rd and 7th and in country it's the root and 7th(Blues too).
A scale generally has 5 to 7 notes. If you think in terms of scales you more than likely not be focusing on those more important tones because you see a scale as simply a collection of notes. If you can see the chords in the scales and see them and the scale change when the harmony does then you'll be doing much better. Learning rhythm "first" will help this.
But if you do not learn your scales and do not focusing on soloing you may develop the opposite problem I have described above. Seeing the Chords too much and not the scales in them. In this case your "solo's" may sound way to much like rhythm because your not able to add non-chord tones to make it more interesting and melodic(melody tends to have a lot of step-wise movement).
Just about any fingering picking tune is based on these principles. This is because they have to provide the harmony, rhythm, and melody at the same time. When you learn these songs you immediately can see how they are doing this. They generally play chords and use the right hand to pick out the melody from the chord and use the fingers of the left hand to add melodic notes not in the chord when they can.
Then again, if rhythm is very easy for you and learning progressions and such are easy then maybe focusing on scales will be more beneficial. The ultimate goal is not to think of chords and scales but to allow the music to flow through unimpeded by conscious thought. One thing you don't want to do is get lost on a progression and just ramble in the correct scale. If you have a strong since of rhythm you'll more likely just go to playing it and possibly adding some simple chord fill that will probably sound good and no one will know you got lost because it fit exactly with the harmony.
If I were just starting out I would spend about 80% on rhythm and 20% on lead. Once I was able to play just about any rhythm and in simple chordal fills(playing notes not in the chord in a melodic like way while still playing the chord) I would change it to about 65% rhythm. I would then work on making the my rhythm sound as good as I can get it and at some point later I would bring it to about 50/50 and possibly start focusing more on soloing.