When muscles contract is the process similar to how non-newtonian fluids react?
Calcium ($\ce{Ca^{2+}}$) is a signalling molecule and it acts like a chemical switch that tells other chemicals in the cell to enact the contraction. It's not accurate to think of the cell "filling up" with $\ce{Ca^{2+}}$ since $\ce{Ca^{2+}}$ wouldn't significantly change the volume of the cell alone *. Also, during contraction, the muscle cells contract in on themselves - they don't swell out.

It's also not accurate to think of the cells as hollow. Other chemicals inside the cell (such as actin and myosin) act like machinery that enact the contraction. Instead of thinking of the cell filling up like a balloon, I think it would be better to think of it containing a chemical winch (as in the figure below) whose pulling force causes the contraction. Similarly, instead of thinking of $\ce{Ca^{2+}}$ as a fluid, think of it as a bunch of particles that cause a chemical reaction to turn on this winch when they're allowed into the cell. 





Reference: https://www.ncbi.nlm.nih.gov/books/NBK21670/

* During a $\ce{Ca^{2+}}$ spike, intracellular $\ce{Ca^{2+}}$ rises around 10-100x from 100 nM. Assuming $\ce{Ca^{2+}}$ has a 100 pm ionic radius, the volume of the cell ($x$) is only increased by around $10^{-9}x$ to $10^{-8}x$, which is negligible.