Firstly all these shapes are squares as they have all the properties that a square has, four equal sides, four right angles.
As there are two different types of paper this problem is on i will split up this solution
ISOMETRIC GRID
On isometric paper you cannot start with any two dots because of the different distances between dots at different angles. For when it is vertical, between two dots there is a gap that is just a bit too small to fit in another dot and still follow the pattern.
Any shapes that do work on this paper have a line of dots through a side. This makes it very easy to find the area as you just square the amount of gaps in the side.
Other possible regular polygons that can be drawn:
Triangle
Hexagon
Rectangle
Other shapes can be drawn but they are not regular
SQUARED GRID
On squared paper you can start with any two dots because of the squares and how they are in neat rows and columns. This means that any line can be transformed into another space on the grid. When they are rotated they can still be placed on the grid. All these aspects can make up a square.
I have tried to find a formula to find the area of the squares but with no luck. I have come to the conclusion that the only way to find it is to measure the sides and square it, as you would with a normal square.
Other possible regular polygons that can be drawn:
Rectangle
Other shapes can be drawn but they are not regular