A major chord is composed of a major 3rd and a minor 3rd. So you'd need a scale where 2 and 4 are a major third apart, and 4 and 6 are a minor third apart.
So you can have: b2-4-b6 or 2-#4-6. So the II will be major using a Locrian, Phrygian, or Lydian scale.
A minor chord is a minor 3rd and a major 3rd. So if the scale has b7-b2-4 or 7-2-#4. Again, it's the Locrian, Phrygian, and Lydian.
None of those scales is non-diatonic, so the answer to the last part is a definite no.
EDIT for the comment: All modes are scales, but almost all scales (with the exception of the whole tone and chromatic scale) are also modes, because you will have a different arrangement of intervals between the tones.
Any scale that has b2-4-b6 or 2-#4-6 will have a major chord on the II. So you could include things like the double harmonic scale, or the major phrygian. You could include non-heptatonic scales like the In scale from Japan. The only real criteria is that it contain the given pitches.
And any scale that has b2-4-b7 or 2-#4-7 will have a minor chord on the 7, like the Japanese Insen scale.
Of course, "7" is now a relative thing, because it could actually be the 5th or 9th or 11th note of the scale. And the harmony would no longer be Tertian. But if you're ok with those tweaks, there will be a lot of scales that will meet your criteria.