TL;DR: It is not unconstitutional to prohibit false statements in commercial speech.
This section is part of the Food and Agrictulture code (FUC) Division 12, Part 4, Chapter 1.
Chapter 1 starts with Article 1, definitions, which includes the section 27501:
Unless the context otherwise requires, the definitions set forth in this article shall govern the construction of this chapter.
and section 27521:
The purposes of this chapter are as follows:
(a) To assure that healthful and wholesome eggs of known quality are sold in this state.
(b) To facilitate the orderly marketing of shell eggs in a uniform manner.
(c) To prevent the marketing of deceptive or mislabeled containers of eggs.
The section 27637 of the chapter must be interpreted in line with this statement of purpose.
With that purpose in mind, the State has a clear reason to restrict false/deceptive/misleading speech when it comes to eggs in the market.
It doesn't restrict speech about eggs in general, only in the context provided by Section 27521. This is not the only legal restriction on false or misleading advertisement and sales practices, there are multitude examples. The Wikipedia provides some references and examples of additional restrictions on commercial speech and relevant precedents.
just be wrong about eggs in California and you can be charged with... something.
If you're making a deceptive/false/misleading marketing statement with regards to eggs, then see section 27581.2: you can be charged with a misdemeanor (under certain conditions, e.g.: repeat offenders), or a civil action (to correct the statement, for example).