Is the statement: "A covenant between me and Allah the Almighty that I will not dye my eyebrows nor trim them" considered a covenant, and can expiation be made for it if it results in hardship, and what is its expiation?

Dyeing hair is permissible for both men and women, provided it is not done with pure black dye, which is forbidden. As for thinning eyebrows to remove a disfigurement, it is permissible if the flaw is considered so by common custom.

If you have made a covenant with Allah to abandon something permissible, then this is an oath, and you may break it and offer expiation for an oath. However, if you have made a covenant with Allah to abandon something forbidden, then you are not permitted to break it, and doing so would be a sin, for which you must offer expiation for an oath.