Another annotation describes Squamous cells as being "identified histologically by the fact that they look flattened and thin..."
This would be describing their physical aspect - like a flat piece of paper.
Flattened in the sense of stale usually refers to taste, tone, or quality of something. Flat beer (lacking effervescence), flat singing, or even a flat mood. This is one of those words that never quite draws a clear distinction however, since flat beer lacks texture, flat singing might lack scale, and a flat mood is often indicated by a flat expression on ones face - each of which suggests something physically flat as well!