Does $\sqsubset$ have any special meaning?

As far as I know, there is no universally accepted meaning for $\sqsubset$ or $\sqsubseteq$. If you see it in a book or article, it will have to be defined by the author in-context.

In general, there are a ton of symbols available in LaTeX (e.g. $\precsim$, $\oplus$, $\curlyvee$) that don't have well-agreed-upon meanings. These are there so that authors have access to plenty of characters to define their own operators.

The square subset symbol is sometimes used to indicate a prefix, so that $x \sqsubseteq y$ denotes that $x$ is a prefix of $y$. This defines a binary relation on strings, called the prefix relation, which is a particular kind of prefix order.

This interpretation seems to make sense for the example you cited:

$(E_n)_{n \in \mathbb{N}}$ satisfies $\forall i \in \mathbb{N}, E_i \sqsubseteq E_{i+1}$

meaning $E_i$ is a prefix of $E_{i+1}$.