Use of colon or other options?
Different versions (or discussions) of Hamlet provide multiple options for sentences structured like the one you present here. Here are 24 alternatives that the first 10 pages of a Google Books search for "to be or not to be that is the question" turned up.
From The Plays of William Shakespeare (1765):
Hamlet. To be, or not to be ? that is the question.--
From The Plays of William Shakspeare (1793):
HAMLET. To be, or not to be, that is the question:—
From William Scott, Lessons in Elocution (1823):
To be—or not to be—that is the question ;
From Edward Bulwer Lytton, "The Present Crisis: A Letter to a Late Cabinet Minister" (1834):
To be, or not to be—that is the question.
From Punch, or the London Charivari, (1843):
To be, or not to be; that is the question.
From Goold Brown, The Grammar of English Grammars: With an Introduction, Historical and Critical (1851), citing Enfield's Sp., p. 367 & Kirkham's Eloc., 123.:
To be, or not to be?—that is the question.
From the same [citing Singer's Shak., ii, 488]:
To be or not to be, that is the question:
From the same [citing Ward's Gram., p. 160]:
To be, or not to be: that is the Question.
From the same [citing Brightland's Gram., p. 209]:
To be, or not to be, that is the Question:
From the same [citing Pinneo's Gram., p. 176]:
To be or not to be! That is the question.
From the same [citing Burgh's Speaker, p. 179]:
To be—or not to be—that is the question.
From the same [the corrected form, according to Mr. Brown]:
To be, or not to be;—that is the question.
From A New Variorum Edition of Shakespeare: Hamlet (1877):
Hamlet. To be or not to be,—that is the question;
From William Wright, "Hamlet," in The Atlantic Monthly (1902):
"To be or not to be," that is "the question."
From Xunwu Chen, Being and Authenticity (2004):
to be or not to be, that is the question!
From Allen Carlson, Aesthetics and the Environment: The Appreciation of Nature, Art and Architecture (2005):
To be, or not to be: that is the question.
From Judith Wright, The One Decision: Make the Single Choice That Will Lead to a Life of More (2006):
To be, or not to be. That is the question.
From John Rowell, To Give Or Not to Give?: Rethinking Dependency, Restoring Generosity, and Redefining Sustainability (2009):
To be, or not to be: that is the question:
From Nilton Bender, To Have or Not to Have That Is the Question: The Economics of Desire (2010):
To be or not to be? That is the question!
From Lisa Nielsen & Willyn Webb, Teaching Generation Text: Using Cell Phones to Enhance Learning (2011):
To be or not to be, that is the question?
From Esther Rashkin, Family Secrets and the Psychoanalysis of Narrative (2014):
“To be or not to be"—that is the question for Hamlet.
From P.I. Sabbathi, Soulzer: Be a Spiritual-Soldier to Undergird Your Own Soul (2014):
To be or not to be... that is the question.
From Michael Rothan, "61 Minutes": Reflections and Homilies for the Year of Matthew (2014):
to be or not to be (that is the question?)
From Dympna Callaghan, Hamlet: Language and Writing (2015):
To be, or not to be — that is the question.
Not all of these formulations are especially good. In particular, the ones suggested by Nielsen & Webb (2011) and Rothan (2014) verge on the ridiculous. But most of the others are coherent enough, and they show considerable range in the punctuation they use to separate the two main phrases of the quotation—comma, semicolon, colon, period, em dash, ellipsis points, question mark, and exclamation point, with quotation marks as another variable.
My preference matches Ricky's, as given in a separate answer: I like the colon or the em dash for separation of the first phrase from the second. But the question mark approach that you prefer isn't wrong, and neither (in my view) are the period, the exclamation point, the comma (despite its seeming to approach a comma splice), and even the semicolon and the ellipsis points.
Each form of punctuation alters the tone or mood or tendency of the quotation slightly, so it's important to choose your weapons with care. But again, any of a number of possibilities may be valid in a particular context.
The sentence is correct.
It is a lame parody of the first line of Hamlet's soliloquy in Act III. To be perfectly fair, the punctuation has varied in it over the past 400 years. Today, there are two ways to punctuate it:
To be, or not to be: that is the question:
Or -
To be, or not to be--that is the question:
That comma after "be," indicating the pause between stating the second choice, would not look right in the student's sentence, since two nouns, rather than two verbs, represent the choices.
To summarize:
Breakfast or sleep: that is the question.
That is the correct way to pucntuate it.