To elaborate on Joseph's answer, the class of continuous images of Cantor cubes has a fancy name, they are the so called dyadic spaces. There is a nice result by Haydon that extends the Borsuk--Dugundji theorem: Every Dugundji space is dyadic. (A space $X$ is Dugundji if the conclusion of the Borsuk--Dungundji theorem holds for $X$.)

R. Haydon, On a problem of Pełczyński: Milutin spaces, Dugundji spaces and AE(0-dim), Studia Math. 52 (1974), 23-31.

It is easy to see that the conclusion of the Borsuk--Dugundji theorem fails for $\beta \mathbb{N}$.

To elaborate on Joseph's answer, the class of continuous images of Cantor cubes has a fancy name, they are the so called dyadic spaces. There is a nice result by Haydon: every Dugundji space is dyadic. (A space $X$ is Dugundji if the conclusion of the Borsuk--Dungundji theorem holds for $X$.)

R. Haydon, On a problem of Pełczyński: Milutin spaces, Dugundji spaces and AE(0-dim), Studia Math. 52 (1974), 23-31.

It is easy to see that the conclusion of the Borsuk--Dugundji theorem fails for $\beta \mathbb{N}$ (it is actually a paradigm counter-example).

To elaborate on Joseph's answer, the class of continuous images of Cantor cubes has a fancy name, they are the so called dyadic spaces. There is a nice result by Haydon that extends the Borsuk--Dugundji theorem: Every Dugundji space is dyadic. (A space $X$ is Dugundji if the conclusion of the Borsuk--Dungundji theorem holds for $X$.)

R. Haydon, On a problem of Pełczyński: Milutin spaces, Dugundji spaces and AE(0-dim), Studia Math. 52 (1974), 23-31.

It is easy to see that the conclusion of the Borsuk--Dugundji theorem fails for $\beta \mathbb{N}$.

To elaborate on Joseph's answer, the class of continuous images of Cantor cubes has a fancy name, they are the so called dyadic spaces. There is a nice result by Haydon that extends the Borsuk--Dugundji theorem: Every Dugundji space is dyadic. (A space $X$ is Dugundji if the conclusion of the Borsuk--Dungundji theorem holds for $X$.)

R. Haydon, On a problem of Pełczyński: Milutin spaces, Dugundji spaces and AE(0-dim), Studia Math. 52 (1974), 23-31.

It is easy to see that the conclusion of the Borsuk--Dugundji theorem fails for $\beta \mathbb{N}$.