Short and incomplete answer:
No, you don't. Technically, you can write how you want - the Cucumber engine just calls method based on steps regex, it doesn't deal with the Scenarios organization in files.
Long and deeper answer:
(Paste short answer here).
Additionally, it is necessary to have a better understanding of the purpose of BDD and Page Objects.
Regarding BDD, I would point to Dan North's article and this post of
Aslak Hellesøy and the article 3 misconceptions about BDD.
With the above-mentioned knowledge, one can understand that Page Objects is just a design pattern, completely unrelated to BDD and Cucumber. The techniques are often used together, but one needs always to justify its usage on a given project, not using it as a single silver-bullet tool out-of-the-box.