Yes, string gauge DOES affect intonation. If your bridge saddles are as far back as they can go (or a fixed bridge) and your intonation is still a couple of cents sharp, use a .001 or .002 Thinner gauge string to flatten the intonation. Conversely, use a thicker gauge string to automatically Sharpen intonation without moving the saddle(s).
2nd, don't listen to the hogwash above about wound strings having more mass than an equal diameter (gauge) plain string. The Wound string of the same gauge actually has Less Mass than the plain string. The reason is the core string plus the winding around it leaves space, which lowers the net mass of the same gauge wound string.
To prove it for yourself, Try changing a plain .018 string to a wound .018 and you will instantly see that the wound string is Way flat on intonation.
First check, or intonate as accurately as you can, with a Good Tuner. If you are flat on a fixed acoustic saddle, use a slightly thicker string to sharpen it up, and vice versa.
IOW, you can really fine tune intonation with string gauge. You can also make a cheap acoustic sound really good.
The same goes for moveable saddles on electric guitars. If the saddle is as far back as it can go and you're Still sharp, install a slightly smaller gauge string to flatten it a bit. This is for Tweak Freaks who want everything Dead Nuts On.
Then order yourself a custom gauge string set from Guitarstringsonline or somebody like that for replacements. The custom set will cost the same as off-the-shelf sets and your guitar will sing.
Remember, Thinner flattens, Thicker sharpens. And also remember that it's mathematically impossible to perfectly intonate a fretted instrument at every fret on every string. There will always be some dissonance.
Back before the '50's, almost all G strings (3rd) were wound. When electrics became popular, lead players liked to stretch the strings a lot, causing the very thin windings to break on the frets.
The solution was to go to a plain G string, but this required a lot more intonation. That's why you still see wound G's on acoustics with fixed bridge saddles, and why the saddle of a plain G on an electric is much farther back. The plain string of equal gauge has more mass, requiring more compensation.
If you're primarily a lead player and you stretch your strings, use the plain string and accept the intonation and tone penalty. If you're primarily a rhythm player, use a wound G for better tone and intonation. If you're using an acoustic with a fixed bridge saddle, definitely use a wound G.
If you change your electric from a plain to a wound G string you'll need to re-intonate the G saddle by moving it forward almost to the B string saddle. The smallest wound strings are .017 and .018 for electrics, from Ball or Daddario. You don't want to go smaller than that anyway because the outer windings are too thin and fragile.
But once intonated, your wound G will be much better in tune up the neck and the tone will be great. Full and partial chords will be sweet.
Daddario has a regular gauge set with an .018 wound 3rd, EXL 110W, or you can buy individual strings and use your normal gauge set for the rest. -Or just buy a custom set online for the same price.
As you might guess, the whole string gauge/type thing gets really involved with a 12 string. That's also another story. Another tweak requires nut compensation, which is also a whole 'nother ball game in itself. Enjoy.