rectangle or square
Not possible - a flat shape with a minimum of four sides must have at least two angles of 90 degrees or larger.. Unless you're allowing 3-D shapes, in which case, a triangular-based pyramid would have four sides, but the angles at the corners would all be less than a right-angle.
A rhombus has 4 equal sides. Its angles can be right angles, but don't have to be. A square is a type of rhombus.
A square has four perpendicular sides.
An octagon has 8 sides and 8 angles.
It is impossible to have a six sided shape with right angles since the lines would never meet, therefore it would not create a shape. The most right angles you can have in a shape is four. A six sided shape is called a hexagon. There are Regular Hexagons, meaning all the sides and all the angles are equal. Any other Hexagon with unequal sides and angles and called Irregular Hexagons. Each side must be straight, and the sum of the angles equaling 720°.
Both statements are not always true. A square is a rectangle (a shape four sides and four corners which are all right angles) but a rectangle is not necessarily a square (a shape with four equal sides and four corners that are all right angles).
A shape with eight sides is always an octagon. If all angles and sides are equal, it is called a regular octagon.
If, by square corners, you mean corners that are right angles, then any polygon with four or more sides can have two square corners. Furthermore, apart from near the corners, the lines need even be straight so the shape need not be a polygon. In the limit a semicircle has two right angles.
A better name for "corners" would be angles, but the shape's name would be a hexagon
A shape with five sides would be a pentagon, but a shape can not have 5 right angles. Squares and Rectangles are both shapes with 4 sides and 4 right angles. They are the only shape with the same amount of right angles as there is sides.
A shape with parallel sides but no right angles is called a parallelogram
A 4 sided polygon is called a quadrilateral. Depending on where the two right angles are, would put the quadrilateral into different subsets. For example if the two right angles are on adjacent corners (and the other 2 angles are not right angles), then it is a trapezoid. If the 2 right angles are in opposite corners, and the other 2 angles are not right angles then it would be a kite-shaped quadrilateral.
A RECTANGLE has 4 sides and 4 right angles.
This shape has only one pair of opposite sides and no right angles: <
A QUADRILATERAL has four corners and four sides, along with four angles. These do not need to be congruent (equal) but they can be.
a pyramid A sqaure, unless you mean three different angles
A Triangle drawn accross a sphere; NOT on a plane surface.