Can an equilateral triangle ever be a right triangle too?

No, it never can be a right triangle.

First, all the angles of an equilateral triangle is 60 degrees because all angles are the same and all the angles add up to 180:

180/3 = 60.

So if it is a right angle it needs one angle 90 degrees which an equilateral triangle can't have,

No, because a equilateral triangle has all even sides and a right triangle has no equal sides. There for a equilateral triangle can not be equal to a right triangle.