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The syllabus for my course requires I draw protein polymers like this, where the patterned boxes represent different bodies (note that this is an extremely simplified drawing stripped down to its bare minimum for high-school students):
Is there any particular reason why the monomers alternate in direction — i.e. up, down and then up?
In this simplified way to draw the primary structure of a protein, there is no reason to enforce flipping orientation of the peptide bonds with each amino acid.
The image you show in your question was probably simply made by taking a more complete drawing of a polypeptide chain similar to the one shown below and then replacing the amino acid center parts with boxes.
Image source: https://en.m.wikipedia.org/wiki/Peptide#/media/File%3ATetrapeptide_structural_formulae_v.1.png
In a Lewis structure, you are free to draw any bond angles and sketch any conformation, no matter whether it matches real bond angles and common conformations.
In the way the polypeptide is drawn in the figure posted by the OP, however, it resembles the conformation found in beta strands forming beta sheets. This type of secondary structure has the polypeptide maximally extended, with consecutive $\ce{CO}$ groups pointing in alternate directions, allowing for hydrogen bonding with an adjacent strand. The two relevant torsion angles, phi and psi, are approximately 180 degrees.
The animation shows an example of a strand that is part of a twisted beta sheet in the structure of the protein LysC:
Source: https://commons.wikimedia.org/wiki/File:Beta_strand.gif
The point where the animation pauses corresponds roughly to the 2D image, with CO-groups (gray/red lines) pointing in alternate direction and in the opposite direction of the NH-groups attached (blue/white lines).
The conformation where CO-groups are opposite of NH-groups of the same peptide bond is conventionally (and slightly inaccurately) called trans peptide. Trans peptides are about a hundred times more common than cis peptides. So it makes sense to draw the Lewis structure hinting at the trans conformation.
There are multiple low energy conformations of peptides (including that of a beta strand) specified in the so-called Ramachandran plot. The other very common comformation features consecutive CO-groups pointing in the same direction, as found in the secondary structure called alpha helix. Drawing an alpha helix in 2D, however, is much less straight forward then drawing the extended conformation of a beta strand.
It has 2 reasons, but not really critical ones:
