Scale in the $y$-direction until we get a circle, divide, scale back. Given the ellipse, if the division of the circle can be done with Euclidean tools, so can the division of the ellipse. If the construction must be done with straightedge and compass, this construction unfortunately excludes almost all odd numbers, though $3$ and $5$ are fine, since the equilateral triangle and the regular pentagon are Euclidean constructible.
Here is some detail. Suppose, for example, that the ellipse has equation $x^2/a^2+y^2/b^2=1$. Draw the circle with center the origin and radius $a$. Let $P$ be one of the division points of the circle. Drop a line from $P$ parallel to the $y$-axis until it meets the ellipse.
Whatever methods of construction are allowed, as long as they work for the circle and include the Euclidean tools, they can be extended to the ellipse.
Comment:: The general idea is fairly powerful. For example, suppose that we want the tangent lines to a standard ellipse that pass through a given point on the $x$-axis. Transform the ellipse to a circle by scaling in the $y$-direction, solve the problem for the circle using geometric ideas, scale back. No calculus needed. Or a simpler problem, find the area of the ellipse with major and minor axes of length $2a$ and $2b$. Using the transform technique, as long as we think we know what scaling in the $y$-direction does to areas, we can conclude that the area of the ellipse is $\pi ab$.