Capitalization with quotation marks when the quote is not direct speech

As Edwin Ashworth says in his comment above, this is a tricky area. Since the two messages that are currently embedded in quotation marks in your example are complete thoughts, I would be inclined to retain the capital at Everybody and Do, regardless of whether I retained the quotation marks or switched to italics for the messages or simply removed the quotation marks. But using either quotation marks or italics suffices to separate the two messages from the rest of the sentence, whether you retain sentence case for the messages or switch to all-lowercase for them.

For what it's worth, The Chicago Manual of Style, fifteenth edition (2003), seems to echo Jane Straus's advice with regard to internal dialogue:

11.47 Unspoken discourse. Thought, imagined dialogue, and other interior discourse may be enclosed in quotation marks or not, according to the context or the writer's preference.

[Example:] "I don't care if we have offended Morgenstern," thought Vera. "Besides," she told herself, "they're all fools."

[Example:] Why, we wondered, did we choose this route?

Unless you are required to follow a more restrictive house style, I think you should feel free to choose your favorite from among (at least) the following five options, each of which sufficiently distinguishes the message text from the surrounding text of the sentence:

We recommend that parents consciously convey the message, "Everybody needs to follow the safety rules" rather than "Do what the babysitter says."

We recommend that parents consciously convey the message, "everybody needs to follow the safety rules" rather than "do what the babysitter says."

We recommend that parents consciously convey the message, Everybody needs to follow the safety rules, rather than Do what the babysitter says.

We recommend that parents consciously convey the message, Everybody needs to follow the safety rules rather than Do what the babysitter says.

We recommend that parents consciously convey the message, everybody needs to follow the safety rules rather than do what the babysitter says.

There may well be other punctuation and typographical treatments that accomplish the same thing.