2,2,4-trimethylpentane will have the highest melting temperature. As can be seen by the space-fill models 2,3,4-trimethylpentane is somewhat flat and planar.
2,2,4-trimethylpentane is slightly more spherical and as expected packs more tightly in the solid crystal structure.

Indeed 2,2,4-trimethylpentane has a melting temperature of -107 ^oC when checking the literature as opposed to -110 ^oC for 2,3,4-trimethylpentane.

As for boiling point 2,3,4-trimethylpentane will have the highest boiling point as it is more planar and thus has a greater surface area exposed to other molecules. Indeed 2,2,4-trimethylpentane has a boiling temperature of only 99 ^oC, when checking the literature, as opposed to 113 ^oC for 2,3,4-trimethylpentane.

2,2,4-trimethylpentane

2,3,4-trimethylpentane