Not with a grid based on regular hexagons
Is it possible to mark squares on a hex map with half hexes?
No because the ratio between the length of the short diagonals and long diagonals (that is: horizontal and vertical distance) in a regular hexagon is irrational. That means no matter how small you dice them, you can never use them to approximate a square.
Or could I layer a square grid over as a measure reference?
No because hex grids and square grids in a tabletop game have different anisotropy. This may be system specific, but in DnD5E on a square grid it is very efficient to walk in multiples of a 45° angle but not very efficient to walk in multiples a 90° angle. On a hex grid, you can only walk in multiples of 60° angles and they are all equally efficient. It gets more complicated if you go more than 1 "step".
Would it be possible to blend the two types of grids? Like hexes used for inside structures & squares for outside?
Technically, no because for the same two reasons above, they will never align properly (at least not at the corners). However, you can of course just do it anyways and accept that the solution isn't perfect.
An irregular hexagon grid can help
If you "smush" the hexagon grid such that the long diagonals in the 90° angle are the same length as the short diagonals in the 0° angle, then they align well with a square grid. You will be able to neatly overlay the grids, compare horizontal and vertical distances, and neatly align grid transitions. It still won't help you with comparing distances in any other direction (e.g. spell ranges) due to the anisotropy problem, though.