how to prove that TE modes cannot exist in microstrip structures
Looks like the image is cut from some microstrip resonator tutorial. The metal surfaces are in YZ planes, their distance h is in X-direction in this image.
Waves which propagate in the insulation layer  must have E perpendicular with the metal surfaces except in case h is at least a half of the wavelength. So thick microstrips work unpredictably, so only Ex can be non-zero. Magnetic field is in XY-plane. If we tried somehow to feed a signal whic has non-zero Ez or Ey those components would be shorted by the metal surfaces. If h were at least half of the wavelength zigzag reflection between the metal surfaces would be possible, but that possibility is excluded to keep the behaviour predictable.
These things are actually the same as in TEM waves which propagate along microstrip transmission lines. The propagation of the wave is well predictable when the Y-width of the line is small, only line length in Z-direction can be a substantial part of the wavelength or several wavelengths.
Resonators can have waves propagating as well to Y and Z directions. Those waves can have both Y and Z magnetic field components. Hx=0. The name of the class of the possible waveforms is settled to &quot;TMx&quot; which obviously is a successful choice. It recalls that this is not a stripline transmission line, but resonator and can have waves which propagate in YZ plane - a difference to TEM waves which propagate to Z direction.
Why there's no TE waves in microstrip resonator? Answer: The possible waves have already got a name - TMx covers them all as long as h is less than half of the wavelength.  Nobody has wanted to spoil his name in publicity by presenting equations for impossible waves and inventing name TE for them.
About boundary conditions:
This is not at all clear. Actually the metal planes and the insulation layers are a complex boundary structure. Practical engineers make simplifications such as the fields do not bulge out of the gap between the metals. To stop YZ plane magnetic field in our thoughts we assume that there's H=0 and Ey=Ez=0 at the edges. In some texts they write it &quot;magnetic wall&quot;
WARNING: Metal tube waveguides have TE waveforms. If you see somewhere text &quot;TE or TM waveform&quot; do not assume anything before you also know the structure geometry. TM in a stripline resonator is totally different thing than TM in metal tube waveguides.