Does mixing calcium dichloride and magnesium sulfate induce a reaction, and how does it affect the hygroscopy of the sample?
You are correct.

In the system $\ce{Ca^2+*Mg^2+*Cl^-*SO4^2-*H2O}$, the least soluble phase is $\ce{CaSO4*2H2O}$.

Once you're mixing $\ce{CaCl2*2H2O}$ and $\ce{MgSO4*7H2O}$, your $\ce{CaCl2*2H2O}$ starts absorbing atmospheric moisture and then dissolving in it. Then you have a brine which dissolves the $\ce{MgSO4*7H2O}$, speeding up the kinetics of the thing. The solid that you precipitate by the recation has less $\ce{H2O}$ per mole of salt, so the excess $\ce{H2O}$ just ends up in the brine (the "water"). Pressing it speeds it up because there is more surface area of the two salts in contact, which is where the reaction happens.